有理标准形

12.2 THE RATIONAL CANONICAL FORM

12.2 有理标准形

We now apply our results on finitely generated modules in the special case where the P.I.D. is the ring $F\left\lbrack x\right\rbrack$ of polynomials in $x$ with coefficients in a field $F$ .

我们现在将我们对有限生成模的结果应用于特殊情况，其中P.I.D.是$F\left\lbrack x\right\rbrack$——带有在域$F$中的系数的$x$多项式环。

令 $V$ 是一个定义在 $F$ 上的有限维向量空间，其维度为 $n$，并且令 $T$ 是 $V$ 的一个固定线性变换（即从 $V$ 到其自身的变换）。正如我们在第10章中看到的，我们可以将 $V$ 视为一个 $F\left\lbrack x\right\rbrack$ -模，其中元素 $x$ 作用于 $V$ 的方式是线性变换 $T$（因此任何 $x$ 的多项式在 $V$ 上的作用都是相同的 $T$ 的多项式）。由于假设 $V$ 在 $F$ 上具有有限维度，因此按照定义，它作为 $F$ -模是有限生成的，因此也必然作为 $F\left\lbrack x\right\rbrack$ -模是有限生成的，所以上一节的分类定理适用。

任何非零自由 $F\left\lbrack x\right\rbrack$ -模（同构于 $F\left\lbrack x\right\rbrack$ 的副本的直接和）是 $F$ 上的无穷维向量空间，所以如果 $V$ 在 $F$ 上有有限维数，那么它实际上必定是一个扭 $F\left\lbrack x\right\rbrack$ -模（即，它的自由秩为 0）。由基本定理可知，$V$ 作为 $F\left\lbrack x\right\rbrack$ -模同构于循环、扭 $F\left\lbrack x\right\rbrack$ -模的直接和。我们将看到这种 $V$ 的分解将允许我们选择一个基，相对于该基，线性变换 $T$ 的矩阵表示形式是特定的简单形式。当我们使用 $V$ 的不变因子分解时，我们得到矩阵 $T$ 的有理标准形，我们在本节中分析它。当我们使用初等因子分解（并且 $F$ 包含 $T$ 的所有特征值时），我们得到约当标准形，这在下一节中讨论，并且之前提到过，它是尽可能接近对角矩阵的 $T$ 的矩阵表示。基本定理的唯一性部分确保了有理标准形和约当标准形的唯一性（这就是为什么它们被称为标准形）。

One important use of these canonical forms is to classify the distinct linear transformations of $V$ . In particular they allow us to determine when two matrices represent the same linear transformation,i.e.,when two given $n \times n$ matrices are similar.

这些标准形的一个重要应用是分类 $V$ 的不同线性变换。特别是它们允许我们确定两个矩阵是否代表相同的线性变换，即两个给定的 $n \times n$ 矩阵何时相似。

Note that this will be another instance where the structure of the space being acted upon (the invariant factor decomposition of $V$ for example) is used to obtain significant information on the algebraic objects (in this case the linear transformations) which are acting. This will be considered in the case of groups acting on vector spaces in Chapter 18 (and goes under the name of Representation Theory of Groups).

注意，这将是另一个例子，其中一个作用于其上的空间的结构（例如 $V$ 的不变因子分解）被用来获取关于代数对象（在这种情况下是线性变换）的显著信息。这将在第18章中考虑群作用于向量空间的情形（并以群表示论的名字出现）。

Before describing the rational canonical form in detail we first introduce some linear algebra.

在详细描述有理标准形之前，我们首先介绍一些线性代数知识。

Definition.

定义。

(1) An element $\lambda$ of $F$ is called an eigenvalue of the linear transformation $T$ if there is a nonzero vector $v \in V$ such that $T\left( v\right) = {\lambda v}$ . In this situation $v$ is called an eigenvector of $T$ with corresponding eigenvalue $\lambda$ .

（1）如果存在一个非零向量 $v \in V$ 使得 $T\left( v\right) = {\lambda v}$ ，则称 $F$ 中的元素 $\lambda$ 为线性变换 $T$ 的特征值。在这种情况下，$v$ 被称为对应于特征值 $\lambda$ 的 $T$ 的特征向量。

(2) If $A$ is an $n \times n$ matrix with coefficients in $F$ ,an element $\lambda$ is called an eigenvalue of $A$ with corresponding eigenvector $v$ if $v$ is a nonzero $n \times 1$ column vector such that ${Av} = {\lambda v}$ .

（2）如果 $A$ 是一个在 $F$ 中具有系数的 $n \times n$ 矩阵，那么如果 $v$ 是一个非零的 $n \times 1$ 列向量，且满足 ${Av} = {\lambda v}$ ，则称元素 $\lambda$ 是 $A$ 的特征值，对应的特征向量为 $v$。

(3) If $\lambda$ is an eigenvalue of the linear transformation $T$ ,the set $\{ v \in V \mid T\left( v\right) = {\lambda v}\}$ is called the eigenspace of $T$ corresponding to the eigenvalue $\lambda$ . Similarly,if $\lambda$ is an eigenvalue of the $n \times n$ matrix $A$ ,the set of $n \times 1$ matrices $v$ with ${Av} = {\lambda v}$ is called the eigenspace of $A$ corresponding to the eigenvalue $\lambda$ .

（3）如果 $\lambda$ 是线性变换 $T$ 的特征值，那么集合 $\{ v \in V \mid T\left( v\right) = {\lambda v}\}$ 被称为对应于特征值 $\lambda$ 的 $T$ 的特征空间。类似地，如果 $\lambda$ 是 $n \times n$ 矩阵 $A$ 的特征值，那么满足 ${Av} = {\lambda v}$ 的 $n \times 1$ 矩阵的集合 $v$ 被称为对应于特征值 $\lambda$ 的 $A$ 的特征空间。

Note that if we fix a basis $\mathcal{B}$ of $V$ then any linear transformation $T$ of $V$ has an associated $n \times n$ matrix $A$ . Conversely,if $A$ is any $n \times n$ matrix then the map $T$ defined by $T\left( v\right) = {Av}$ for $v \in V$ ,where the $v$ on the right is the $n \times 1$ vector consisting of the coordinates of $v$ with respect to the fixed basis $\mathcal{B}$ of $V$ ,is a linear transformation of $V$ . Then $v$ is an eigenvector of $T$ with corresponding eigenvalue $\lambda$ if and only if the coordinate vector of $v$ with respect to $\mathcal{B}$ is an eigenvector of $A$ with eigenvalue $\lambda$ . In other words,the eigenvalues for the linear transformation $T$ are the same as the eigenvalues for the matrix $A$ of $T$ with respect to any fixed basis for $V$ .

请注意，如果我们固定 $\mathcal{B}$ 的一个基 $V$，那么任何 $V$ 的线性变换 $T$ 都有一个关联的 $n \times n$ 矩阵 $A$。反之，如果 $A$ 是任意的 $n \times n$ 矩阵，那么由 $T\left( v\right) = {Av}$ 定义的映射 $T$ 对于 $v \in V$，其中右侧的 $v$ 是相对于固定基 $\mathcal{B}$ 的 $V$ 中 $v$ 坐标组成的 $n \times 1$ 向量，是一个 $V$ 的线性变换。当且仅当相对于 $\mathcal{B}$ 的 $v$ 坐标向量是 $A$ 的特征向量，对应的特征值为 $\lambda$ 时，$v$ 是 $T$ 的特征向量。换句话说，线性变换 $T$ 的特征值与任何固定基对于 $V$ 的 $A$ 矩阵的特征值相同。

Definition. The determinant of a linear transformation from $V$ to $V$ is the determinant of any matrix representing the linear transformation (note that this does not depend on the choice of the basis used).

定义。从 $V$ 到 $V$ 的线性变换的行列式是表示该线性变换的任意矩阵的行列式（注意，这并不依赖于所使用的基的选择）。

Proposition 12. The following are equivalent:

命题 12。以下等价：

(1) $\lambda$ is an eigenvalue of $T$

(1) $\lambda$ 是 $T$ 的特征值

(2) ${\lambda I} - T$ is a singular linear transformation of $V$

(2) ${\lambda I} - T$ 是 $V$ 的奇异线性变换

(3) $\det \left( {{\lambda I} - T}\right) = 0$ .

(3) $\det \left( {{\lambda I} - T}\right) = 0$。

Proof: Since $\lambda$ is an eigenvalue of $T$ with corresponding eigenvector $v$ if and only if $v$ is a nonzero vector in the kernel of ${\lambda I} - T$ ,it follows that (1) and (2) are equivalent. (2) and (3) are equivalent by our results on determinants.

证明：由于 $\lambda$ 是 $T$ 的特征值，对应的特征向量为 $v$，当且仅当 $v$ 是 ${\lambda I} - T$ 核中的非零向量，因此 (1) 和 (2) 等价。根据我们对行列式的研究结果，(2) 和 (3) 也是等价的。

Definition. Let $x$ be an indeterminate over $F$ . The polynomial $\det \left( {{xI} - T}\right)$ is called the characteristic polynomial of $T$ and will be denoted ${c}_{T}\left( x\right)$ . If $A$ is an $n \times n$ matrix with coefficients in $F,\det \left( {{xI} - A}\right)$ is called the characteristic polynomial of $A$ and will be denoted ${c}_{A}\left( x\right)$ .

定义。设 $x$ 是 $F$ 上的一个不定元。多项式 $\det \left( {{xI} - T}\right)$ 被称为 $T$ 的特征多项式，并表示为 ${c}_{T}\left( x\right)$。如果 $A$ 是一个在 $F,\det \left( {{xI} - A}\right)$ 中具有系数的 $n \times n$ 矩阵，则该矩阵的特征多项式被称为 $A$ 的特征多项式，并表示为 ${c}_{A}\left( x\right)$。

It is easy to see by expanding the determinant that the characteristic polynomial of either $T$ or $A$ is a monic polynomial of degree $n = \dim V$ . Proposition 12 says that the set of eigenvalues of $T$ (or $A$ ) is precisely the set of roots of the characteristic polynomial of $T$ (of $A$ ,respectively). In particular, $T$ has at most $n$ distinct eigenvalues.

通过展开行列式可以容易地看出，$T$ 或 $A$ 的特征多项式是次数为 $n = \dim V$ 的单变量多项式。命题 12 说明 $T$（或 $A$）的特征值集合恰好是 $T$（或分别的 $A$）特征多项式的根集合。特别是，$T$ 最多有 $n$ 个不同的特征值。

We have seen that $V$ considered as a module over $F\left\lbrack x\right\rbrack$ via the linear transformation $T$ is a torsion $F\left\lbrack x\right\rbrack$ -module. Let $m\left( x\right) \in F\left\lbrack x\right\rbrack$ be the unique monic polynomial generating the annihilator of $V$ in $F\left\lbrack x\right\rbrack$ . Equivalently, $m\left( x\right)$ is the unique monic polynomial of minimal degree annihilating $V$ (i.e.,such that $m\left( T\right)$ is the 0 linear transformation),and if $f\left( x\right) \in F\left\lbrack x\right\rbrack$ is any polynomial annihilating $V,m\left( x\right)$ divides $f\left( x\right)$ . Since the ring of all $n \times n$ matrices over $F$ is isomorphic to the collection of all linear transformations of $V$ to itself (an isomorphism is obtained by choosing a basis for $V$ ),it follows that for any $n \times n$ matrix $A$ over $F$ there is similarly a unique monic polynomial of minimal degree with $m\left( A\right)$ the zero matrix.

我们已经看到，通过线性变换 $T$ 将 $V$ 视为 $F\left\lbrack x\right\rbrack$ 的模是一个扭模。设 $m\left( x\right) \in F\left\lbrack x\right\rbrack$ 是唯一生成 $V$ 在 $F\left\lbrack x\right\rbrack$ 中消灭子的单变量多项式。等价地，$m\left( x\right)$ 是唯一消灭 $V$ 的最小次数的单变量多项式（即，使得 $m\left( T\right)$ 是零线性变换），如果 $f\left( x\right) \in F\left\lbrack x\right\rbrack$ 是任何消灭 $V,m\left( x\right)$ 的多项式，则 $V,m\left( x\right)$ 除 $f\left( x\right)$。由于所有 $n \times n$ 矩阵在 $F$ 上的环同构于所有将 $V$ 映射到自身的线性变换的集合（通过为 $V$ 选择一个基可以得到同构），因此对于任何在 $F$ 上的 $n \times n$ 矩阵 $A$，也存在一个具有 $m\left( A\right)$ 零矩阵的唯一最小次数的单变量多项式。

Definition. The unique monic polynomial which generates the ideal $\operatorname{Ann}\left( V\right)$ in $F\left\lbrack x\right\rbrack$ is called the minimal polynomial of $T$ and will be denoted ${m}_{T}\left( x\right)$ . The unique monic polynomial of smallest degree which when evaluated at the matrix $A$ is the zero matrix is called the minimal polynomial of $A$ and will be denoted ${m}_{A}\left( x\right)$ .

定义。生成理想 $\operatorname{Ann}\left( V\right)$ 在 $F\left\lbrack x\right\rbrack$ 中的唯一单变量多项式称为 $T$ 的最小多项式，并表示为 ${m}_{T}\left( x\right)$ 。当在矩阵 $A$ 上求值时，唯一的最小次数的单变量多项式，其结果为零矩阵，称为 $A$ 的最小多项式，并表示为 ${m}_{A}\left( x\right)$ 。

It is easy to see (cf. Exercise 5) that the degrees of these minimal polynomials are at most ${n}^{2}$ where $n$ is the dimension of $V$ . We shall shortly prove that the minimal polynomial for $T$ is a divisor of the characteristic polynomial for $T$ (this is the Cayley-Hamilton Theorem),and similarly for $A$ ,so in fact the degrees of these polynomials are at most $n$ .

很容易看出（参见练习5），这些最小多项式的次数最多为 ${n}^{2}$，其中 $n$ 是 $V$ 的维度。我们很快将证明 $T$ 的最小多项式是 $T$ 的特征多项式的因子（这是凯莱-哈密顿定理），对于 $A$ 也是如此，因此实际上这些多项式的次数最多为 $n$。

We now describe the rational canonical form of the linear transformation $T$ (respectively,of the $n \times n$ matrix $A$ ). By Theorem 5 we have an isomorphism

我们现在描述线性变换 $T$（分别是 $n \times n$ 矩阵 $A$ 的）的有理标准形。根据定理5，我们有一个同构

of $F\left\lbrack x\right\rbrack$ -modules where ${a}_{1}\left( x\right) ,{a}_{2}\left( x\right) ,\ldots ,{a}_{m}\left( x\right)$ are polynomials in $F\left\lbrack x\right\rbrack$ of degree at least one with the divisibility conditions

的 $F\left\lbrack x\right\rbrack$ -模，其中 ${a}_{1}\left( x\right) ,{a}_{2}\left( x\right) ,\ldots ,{a}_{m}\left( x\right)$ 是次数至少为1的 $F\left\lbrack x\right\rbrack$ 中的多项式，满足整除条件

These invariant factors ${a}_{i}\left( x\right)$ are only determined up to a unit in $F\left\lbrack x\right\rbrack$ but since the units of $F\left\lbrack x\right\rbrack$ are precisely the nonzero elements of $F$ (i.e., the nonzero constant polynomials), we may make these polynomials unique by stipulating that they be monic.

这些不变因子 ${a}_{i}\left( x\right)$ 只在 $F\left\lbrack x\right\rbrack$ 中的单位上确定，但由于 $F\left\lbrack x\right\rbrack$ 的单位正是 $F$ 的非零元素（即非零常数多项式），我们可以通过指定它们为单变量多项式来使这些多项式唯一。

Since the annihilator of $V$ is the ideal $\left( {{a}_{m}\left( x\right) }\right)$ (part (3) of Theorem 5),we immediately obtain:

由于 $V$ 的消灭子是理想 $\left( {{a}_{m}\left( x\right) }\right)$（定理5的第三部分），我们立即得到：

Proposition 13. The minimal polynomial ${m}_{T}\left( x\right)$ is the largest invariant factor of $V$ . All the invariant factors of $V$ divide ${m}_{T}\left( x\right)$ .

命题13。最小多项式 ${m}_{T}\left( x\right)$ 是 $V$ 的最大不变因子。 $V$ 的所有不变因子都整除 ${m}_{T}\left( x\right)$ 。

We shall see below how to calculate not only the minimal polynomial for $T$ but also the other invariant factors.

我们将在下面看到如何计算 $T$ 的最小多项式，以及其他不变因子。

We now choose a basis for each of the direct summands for $V$ in the decomposition (1)above for which the matrix for $T$ is quite simple. Recall that the linear transformation $T$ acting on the left side of (1) is the element $x$ acting by multiplication on each of the factors on the right side of the isomorphism in (1).

我们现在为 $V$ 的每个直和项选择一个基，使得在分解 (1) 中的矩阵 $T$ 非常简单。回想一下，线性变换 $T$ 作用于 (1) 左侧的元素 $x$ 通过乘法作用于同构 (1) 右侧的每个因子。

We have seen in the example following Proposition 1 of Chapter 11 that the elements $1,\bar{x},{\bar{x}}^{2},\ldots ,{\bar{x}}^{k - 1}$ give a basis for the vector space $F\left\lbrack x\right\rbrack /\left( {a\left( x\right) }\right)$ where $a\left( x\right) = {x}^{k} +$ ${b}_{k - 1}{x}^{k - 1} + \cdots + {b}_{1}x + {b}_{0}$ is any monic polynomial in $F\left\lbrack x\right\rbrack$ and $\bar{x} = x{\;\operatorname{mod}\;\left( {a\left( x\right) }\right) }$ . With respect to this basis the linear transformation of multiplication by $x$ acts in a simple manner:

我们在第11章命题1后的例子中看到，元素 $1,\bar{x},{\bar{x}}^{2},\ldots ,{\bar{x}}^{k - 1}$ 构成向量空间 $F\left\lbrack x\right\rbrack /\left( {a\left( x\right) }\right)$ 的一个基，其中 $a\left( x\right) = {x}^{k} +$ ${b}_{k - 1}{x}^{k - 1} + \cdots + {b}_{1}x + {b}_{0}$ 是 $F\left\lbrack x\right\rbrack$ 中的任意首一多项式，$\bar{x} = x{\;\operatorname{mod}\;\left( {a\left( x\right) }\right) }$ 。对于这个基，乘以 $x$ 的线性变换作用方式很简单：

$x :$

where the last equality is because ${\bar{x}}^{k} + {b}_{k - 1}{\bar{x}}^{k - 1} + \cdots + {b}_{1}\bar{x} + {b}_{0} = 0$ since $a\left( \bar{x}\right) = 0$ in $F\left\lbrack x\right\rbrack /\left( {a\left( x\right) }\right)$ . With respect to this basis,the matrix formultiplication by $x$ is therefore

其中最后一个等式是因为 ${\bar{x}}^{k} + {b}_{k - 1}{\bar{x}}^{k - 1} + \cdots + {b}_{1}\bar{x} + {b}_{0} = 0$ ，由于 $a\left( \bar{x}\right) = 0$ 在 $F\left\lbrack x\right\rbrack /\left( {a\left( x\right) }\right)$ 中。对于这个基，乘以 $x$ 的矩阵形式因此是

Such matrices are given a name:

这样的矩阵有一个名称：

Definition. Let $a\left( x\right) = {x}^{k} + {b}_{k - 1}{x}^{k - 1} + \cdots + {b}_{1}x + {b}_{0}$ be any monic polynomial in $F\left\lbrack x\right\rbrack$ . The companion matrix of $a\left( x\right)$ is the $k \times k$ matrix with 1’s down the first subdiagonal, $- {b}_{0}, - {b}_{1},\ldots , - {b}_{k - 1}$ down the last column and zeros elsewhere. The companion matrix of $a\left( x\right)$ will be denoted by ${\mathcal{C}}_{a\left( x\right) }$ .

定义。设 $a\left( x\right) = {x}^{k} + {b}_{k - 1}{x}^{k - 1} + \cdots + {b}_{1}x + {b}_{0}$ 为 $F\left\lbrack x\right\rbrack$ 中的任意首一多项式。$a\left( x\right)$ 的伴随矩阵是具有1沿着第一条次对角线下方，$- {b}_{0}, - {b}_{1},\ldots , - {b}_{k - 1}$ 沿着最后一列，其余位置为0的 $k \times k$ 矩阵。$a\left( x\right)$ 的伴随矩阵将表示为 ${\mathcal{C}}_{a\left( x\right) }$ 。

We apply this to each of the cyclic modules on the right side of (1) above and let ${\mathcal{B}}_{i}$ be the elements of $V$ corresponding to the basis chosen above for the cyclic factor $F\left\lbrack x\right\rbrack /\left( {{a}_{i}\left( x\right) }\right)$ under the isomorphism in (1). Then by definition the linear transformation $T$ acts on ${\mathcal{B}}_{i}$ by the companion matrix for ${a}_{i}\left( x\right)$ since we have seen that this is how multiplication by $x$ acts. The union $\mathcal{B}$ of the ${\mathcal{B}}_{i}$ ’s gives a basis for $V$ since the sum on the right of (1) is direct and with respect to this basis the linear transformation $T$ has as matrix the direct sum of the companion matrices for the invariant factors, i.e.,

我们将此应用于上述（1）右侧的每个循环模块，并让 ${\mathcal{B}}_{i}$ 为 $V$ 中与上述选择的循环因子 $F\left\lbrack x\right\rbrack /\left( {{a}_{i}\left( x\right) }\right)$ 在同构（1）下对应的元素。根据定义，线性变换 $T$ 通过 ${a}_{i}\left( x\right)$ 的伴随矩阵作用于 ${\mathcal{B}}_{i}$，因为我们已经看到这是乘以 $x$ 的作用方式。所有 ${\mathcal{B}}_{i}$ 的并集给出了 $V$ 的一个基，因为（1）右侧的和是直接的，并且对于这个基，线性变换 $T$ 的矩阵是伴随矩阵的直接和，即，

(12.2)

Notice that this matrix is uniquely determined from the invariant factors of the $F\left\lbrack x\right\rbrack$ - module $V$ and,by Theorem 9,the list of invariant factors uniquely determines the module $V$ up to isomorphism as an $F\left\lbrack x\right\rbrack$ -module.

注意，这个矩阵由 $F\left\lbrack x\right\rbrack$ - 模块的不变因子唯一确定，并且，根据定理9，不变因子的列表唯一地确定了作为 $F\left\lbrack x\right\rbrack$ - 模块的 $V$ 的同构。

Definition.

定义。

(1) A matrix is said to be in rational canonical form if it is the direct sum of companion matrices for monic polynomials ${a}_{1}\left( x\right) ,\ldots ,{a}_{m}\left( x\right)$ of degree at least one with ${a}_{1}\left( x\right) \left| {{a}_{2}\left( x\right) }\right| \cdots \mid {a}_{m}\left( x\right)$ . The polynomials ${a}_{i}\left( x\right)$ are called the invariant factors of the matrix. Such a matrix is also said to be a block diagonal matrix with blocks the companion matrices for the ${a}_{i}\left( x\right)$ .

（1）如果一个矩阵是单变量多项式 ${a}_{1}\left( x\right) ,\ldots ,{a}_{m}\left( x\right)$ 的伴随矩阵的直接和，且这些多项式的次数至少为1，并且具有 ${a}_{1}\left( x\right) \left| {{a}_{2}\left( x\right) }\right| \cdots \mid {a}_{m}\left( x\right)$ ，则称这个矩阵为有理标准形。这些多项式 ${a}_{i}\left( x\right)$ 被称为矩阵的不变因子。这样的矩阵也被称为以 ${a}_{i}\left( x\right)$ 的伴随矩阵为块的块对角矩阵。

(2) A rational canonical form for a linear transformation $T$ is a matrix representing $T$ which is in rational canonical form.

（2）一个线性变换 $T$ 的有理标准形是一个表示 $T$ 的矩阵，且该矩阵处于有理标准形。

We have seen that any linear transformation $T$ has a rational canonical form. We now see that this rational canonical form is unique (hence is called the rational canonical form for $T$ ). To see this note that the process we used to determine the matrix of $T$ from the direct sum decomposition is reversible. Suppose ${b}_{1}\left( x\right) ,{b}_{2}\left( x\right) ,\ldots ,{b}_{t}\left( x\right)$ are monic polynomials in $F\left\lbrack x\right\rbrack$ of degree at least one such that ${b}_{i}\left( x\right) \mid {b}_{i + 1}\left( x\right)$ for all $i$ and suppose for some basis $\mathcal{E}$ of $V$ ,that the matrix of $T$ with respect to the basis $\mathcal{E}$ is the direct sum of the companion matrices of the ${b}_{i}\left( x\right)$ . Then $V$ must be a direct sum of $T$ -stable subspaces ${D}_{i}$ ,one for each ${b}_{i}\left( x\right)$ in such a way that the matrix of $T$ on each ${D}_{i}$ is the companion matrix of ${b}_{i}\left( x\right)$ . Let ${\mathcal{E}}_{i}$ be the corresponding (ordered) basis of ${D}_{i}$ (so $\mathcal{E}$ is the union of the ${\mathcal{E}}_{i}$ ) and let ${e}_{i}$ be the first basis element in ${\mathcal{E}}_{i}$ . Then it is easy to see that ${D}_{i}$ is a cyclic $F\left\lbrack x\right\rbrack$ -module with generator ${e}_{i}$ and that the annihilator of ${D}_{i}$ is ${b}_{i}\left( x\right)$ . Thus the torsion $F\left\lbrack x\right\rbrack$ -module $V$ decomposes into a direct sum of cyclic $F\left\lbrack x\right\rbrack$ -modules in two ways, both of which satisfy the conditions of Theorem 5, i.e., both of which give lists of invariant factors. Since the invariant factors are unique by Theorem 9, ${a}_{i}\left( x\right)$ and ${b}_{i}\left( x\right)$ must differ by a unit factor in $F\left\lbrack x\right\rbrack$ and since the polynomials are monic by assumption,we must have ${a}_{i}\left( x\right) = {b}_{i}\left( x\right)$ for all $i$ . This proves the following result:

我们已经看到，任何线性变换 $T$ 都有一个有理标准形。现在我们知道这个有理标准形是唯一的（因此被称为 $T$ 的有理标准形）。为了证明这一点，请注意我们用来从直接和分解确定 $T$ 的矩阵的过程是可逆的。假设 ${b}_{1}\left( x\right) ,{b}_{2}\left( x\right) ,\ldots ,{b}_{t}\left( x\right)$ 是 $F\left\lbrack x\right\rbrack$ 中的首一多项式，且次数至少为一个，使得对于所有 $i$ 都有 ${b}_{i}\left( x\right) \mid {b}_{i + 1}\left( x\right)$，并且假设对于某个 $V$ 的基 $\mathcal{E}$，$T$ 关于基 $\mathcal{E}$ 的矩阵是 ${b}_{i}\left( x\right)$ 的伴随矩阵的直接和。那么 $V$ 必须是 $T$ -稳定子空间的直接和，每个 ${b}_{i}\left( x\right)$ 有一个，使得 $T$ 在每个 ${D}_{i}$ 上的矩阵是 ${b}_{i}\left( x\right)$ 的伴随矩阵。设 ${\mathcal{E}}_{i}$ 是 ${D}_{i}$ 的相应（有序）基（因此 $\mathcal{E}$ 是 ${\mathcal{E}}_{i}$ 的并集），设 ${e}_{i}$ 是 ${\mathcal{E}}_{i}$ 中的第一个基元素。那么很容易看出 ${D}_{i}$ 是一个循环 $F\left\lbrack x\right\rbrack$ -模，生成元为 ${e}_{i}$，并且 ${D}_{i}$ 的消元子为 ${b}_{i}\left( x\right)$。因此，挠 $F\left\lbrack x\right\rbrack$ -模 $V$ 以两种方式分解为循环 $F\left\lbrack x\right\rbrack$ -模的直接和，这两种方式都满足定理 5 的条件，即都给出不变因子的列表。由于定理 9 证明了不变因子是唯一的，${a}_{i}\left( x\right)$ 和 ${b}_{i}\left( x\right)$ 必须在 $F\left\lbrack x\right\rbrack$ 中相差一个单位因子，并且由于假设多项式是首一的，我们必须对于所有 $i$ 有 ${a}_{i}\left( x\right) = {b}_{i}\left( x\right)$。这证明了以下结果：

Theorem 14. (Rational Canonical Form for Linear Transformations) Let $V$ be a finite dimensional vector space over the field $F$ and let $T$ be a linear transformation of $V$ .

定理 14.（线性变换的有理标准形）设 $V$ 是定义在域 $F$ 上的有限维向量空间，设 $T$ 是 $V$ 的一个线性变换。

(1) There is a basis for $V$ with respect to which the matrix for $T$ is in rational canonical form, i.e., is a block diagonal matrix whose diagonal blocks are the companion matrices for monic polynomials ${a}_{1}\left( x\right) ,{a}_{2}\left( x\right) ,\ldots ,{a}_{m}\left( x\right)$ of degree at least one with ${a}_{1}\left( x\right) \left| {{a}_{2}\left( x\right) }\right| \cdots \mid {a}_{m}\left( x\right)$ .

（1）存在 $V$ 的一个基，对于该基，$T$ 的矩阵是有理标准形，即是一个对角块矩阵，其对角块是首一多项式 ${a}_{1}\left( x\right) ,{a}_{2}\left( x\right) ,\ldots ,{a}_{m}\left( x\right)$ 的伴随矩阵，其次数至少为 1 并且 ${a}_{1}\left( x\right) \left| {{a}_{2}\left( x\right) }\right| \cdots \mid {a}_{m}\left( x\right)$。

(2) The rational canonical form for $T$ is unique.

（2）$T$ 的有理标准形是唯一的。

The use of the word rational is to indicate that this canonical form is calculated entirely within the field $F$ and exists for any linear transformation $T$ . This is not the case for the Jordan canonical form (considered later),which only exists if the field $F$ contains the eigenvalues for $T$ (cf. also the remarks following Corollary 18).

使用“有理”一词是为了表明这种标准形完全在域 $F$ 内计算，并且对于任何线性变换 $T$ 都存在。这与约当标准形（稍后讨论）的情况不同，后者仅在域 $F$ 包含 $T$ 的特征值时存在（参见 also the remarks following Corollary 18）。

The following result translates the notion of similar linear transformations (i.e., the same linear transformation up to a change of basis) into the language of modules and relates this notion to rational canonical forms.

下面的结果将相似线性变换（即，在基变换下的相同线性变换）的概念转化为模块的语言，并将此概念与有理标准形联系起来。

Theorem 15. Let $S$ and $T$ be linear transformations of $V$ . Then the following are equivalent:

定理 15. 设 $S$ 和 $T$ 是 $V$ 的线性变换。那么以下条件是等价的：

(1) $S$ and $T$ are similar linear transformations

（1）$S$ 和 $T$ 是相似的线性变换

(2) the $F\left\lbrack x\right\rbrack$ -modules obtained from $V$ via $S$ and via $T$ are isomorphic $F\left\lbrack x\right\rbrack$ - modules

（2）通过 $V$ 和通过 $S$ 从 $F\left\lbrack x\right\rbrack$ 获得的 $F\left\lbrack x\right\rbrack$ -模块与 $T$ 是同构的 $F\left\lbrack x\right\rbrack$ -模块

(3) $S$ and $T$ have the same rational canonical form.

（3）$S$ 和 $T$ 具有相同的有理标准形。

Proof: [(1) implies (2)] Assume there is a nonsingular linear transformation $U$ such that $S = {UT}{U}^{-1}$ . The vector space isomorphism $U : V \rightarrow V$ is also an $F\left\lbrack x\right\rbrack$ -module homomorphism,where $x$ acts on the first $V$ via $T$ and on the second via $S$ ,since for example $U\left( {xv}\right) = U\left( {Tv}\right) = {UT}\left( v\right) = {SU}\left( v\right) = x\left( {Uv}\right)$ . Hence this is an $F\left\lbrack x\right\rbrack$ -module isomorphism of the two modules in (2).

证明：[(1) 蕴含 (2)] 假设存在一个非奇异线性变换 $U$ 使得 $S = {UT}{U}^{-1}$ 。该向量空间同构 $U : V \rightarrow V$ 同时也是一个 $F\left\lbrack x\right\rbrack$ -模同态，其中 $x$ 通过 $V$ 作用于第一个分量，通过 $T$ 作用于第二个分量，因为例如 $S$ 。因此，这是 (2) 中两个模的同构。

[(2) implies (3)] Assume (2) holds and denote by ${V}_{1}$ the vector space $V$ made into an $F\left\lbrack x\right\rbrack$ -module via $S$ and denote by ${V}_{2}$ the space $V$ made into an $F\left\lbrack x\right\rbrack$ -module via $T$ . Since ${V}_{1} \cong {V}_{2}$ as $F\left\lbrack x\right\rbrack$ -modules they have the same list of invariant factors. Thus $S$ and $T$ have a common rational canonical form.

[(2) 蕴含 (3)] 假设 (2) 成立，并记 ${V}_{1}$ 为通过 $S$ 使向量空间 $V$ 成为 $F\left\lbrack x\right\rbrack$ -模，记 ${V}_{2}$ 为通过 $T$ 使空间 $V$ 成为 $F\left\lbrack x\right\rbrack$ -模。由于 ${V}_{1} \cong {V}_{2}$ 作为 $F\left\lbrack x\right\rbrack$ -模，它们具有相同的不变因子列表。因此 $S$ 和 $T$ 有一个共同的理性标准形。

[(3) implies (1)] Assume (3) holds. Since $S$ and $T$ have the same matrix representation with respect to some choice of (possibly different) bases of $V$ by assumption, they are,up to a change of basis,the same linear transformation of $V$ ,hence are similar.

[(3) 蕴含 (1)] 假设 (3) 成立。由于根据假设，$S$ 和 $T$ 在某些（可能是不同的）$V$ 基底下的矩阵表示相同，它们在基底变换下是相同的 $V$ 线性变换，因此是相似的。

Let $A$ be any $n \times n$ matrix with entries from $F$ . Let $V$ be an $n$ -dimensional vector space over $F$ . Recall we can then define a linear transformation $T$ on $V$ by choosing a basis for $V$ and setting $T\left( v\right) = {Av}$ where $v$ on the right hand side means the $n \times 1$ column vector of coordinates of $v$ with respect to our chosen basis (this is just the usual identification of linear transformations with matrices). Then (of course) the matrix for this $T$ with respect to this basis is the given matrix $A$ . Put another way,any $n \times n$ matrix $A$ with entries from the field $F$ arises as the matrix for some linear transformation $T$ of an $n$ -dimensional vector space.

设 $A$ 为任意 $n \times n$ 矩阵，其元素来自 $F$ 。设 $V$ 为 $n$ 维的 $F$ 上的向量空间。回顾我们可以在 $V$ 上定义一个线性变换 $T$ ，通过为 $V$ 选择一个基并设置 $T\left( v\right) = {Av}$ ，其中 $v$ 右侧的 $n \times 1$ 表示 $v$ 关于我们选择的基的坐标的 $n \times 1$ 列向量（这只是线性变换与矩阵的通常识别）。那么（当然）这个基下 $T$ 的矩阵就是给定的矩阵 $A$ 。换句话说，任何来自域 $F$ 的 $n \times n$ 矩阵 $A$ 都可以作为某个 $n$ 维向量空间的线性变换 $T$ 的矩阵。

This dictionary between linear transformations of vector spaces and matrices allows us to state our previous two results in the language of matrices:

这个线性变换的向量空间和矩阵之间的字典允许我们用矩阵的语言陈述我们之前得到的结果：

Theorem 16. (Rational Canonical Form for Matrices) Let $A$ be an $n \times n$ matrix over the field $F$ .

定理 16。（矩阵的有理标准形）设 $A$ 为 $F$ 上的 $n \times n$ 矩阵。

(1) The matrix $A$ is similar to a matrix in rational canonical form,i.e.,there is an invertible $n \times n$ matrix $P$ over $F$ such that ${P}^{-1}{AP}$ is a block diagonal matrix whose diagonal blocks are the companion matrices for monic polynomials ${a}_{1}\left( x\right) ,{a}_{2}\left( x\right) ,\ldots ,{a}_{m}\left( x\right)$ of degree at least one with ${a}_{1}\left( x\right) \left| {{a}_{2}\left( x\right) }\right| \cdots \mid {a}_{m}\left( x\right) .$

（1）矩阵 $A$ 与有理标准形的一个矩阵相似，即存在一个可逆的 $n \times n$ 矩阵 $P$ 在 $F$ 上，使得 ${P}^{-1}{AP}$ 是一个块对角矩阵，其对角块是单变量多项式 ${a}_{1}\left( x\right) ,{a}_{2}\left( x\right) ,\ldots ,{a}_{m}\left( x\right)$ 的伴随矩阵，其次数至少为一个，并且 ${a}_{1}\left( x\right) \left| {{a}_{2}\left( x\right) }\right| \cdots \mid {a}_{m}\left( x\right) .$

(2) The rational canonical form for $A$ is unique.

（2）$A$ 的有理标准形是唯一的。

Definition. The invariant factors of an $n \times n$ matrix over a field $F$ are the invariant factors of its rational canonical form.

定义。一个 $n \times n$ 矩阵在域 $F$ 上的不变因子是其有理标准形的不变因子。

Theorem 17. Let $A$ and $B$ be $n \times n$ matrices over the field $F$ . Then $A$ and $B$ are similar if and only if $A$ and $B$ have the same rational canonical form.

定理 17。设 $A$ 和 $B$ 是定义在域 $F$ 上的 $n \times n$ 矩阵。那么 $A$ 和 $B$ 相似当且仅当它们具有相同的理性标准形。

If $A$ is a matrix with entries from a field $F$ and $F$ is a subfield of a larger field $K$ then we may also consider $A$ as a matrix over $K$ . The next result shows that the rational canonical form for $A$ and questions of similarity do not depend on which field contains the entries of $A$ .

如果 $A$ 是一个元素来自域 $F$ 的矩阵，且 $F$ 是更大域 $K$ 的子域，那么我们也可以将 $A$ 视为定义在 $K$ 上的矩阵。下一个结果说明，$A$ 的理性标准形以及相似性问题不依赖于包含 $A$ 元素的域。

Corollary 18. Let $A$ and $B$ be two $n \times n$ matrices over a field $F$ and suppose $F$ is a subfield of the field $K$ .

推论 18。设 $A$ 和 $B$ 是定义在域 $F$ 上的两个 $n \times n$ 矩阵，并假设 $F$ 是域 $K$ 的子域。

(1) The rational canonical form of $A$ is the same whether it is computed over $K$ or over $F$ . The minimal and characteristic polynomials and the invariant factors of $A$ are the same whether $A$ is considered as a matrix over $F$ or as a matrix over $K$ .

（1）无论在 $K$ 上还是在 $F$ 上计算，$A$ 的理性标准形都是相同的。最小多项式和特征多项式以及 $A$ 的不变因子，无论将 $A$ 视为在 $F$ 上的矩阵还是 $K$ 上的矩阵，都是相同的。

(2) The matrices $A$ and $B$ are similar over $K$ if and only if they are similar over $F$ ,i.e.,there exists an invertible $n \times n$ matrix $P$ with entries from $K$ such that $B = {P}^{-1}{AP}$ if and only if there exists an (in general different) invertible $n \times n$ matrix $Q$ with entries from $F$ such that $B = {Q}^{-1}{AQ}$ .

（2）如果矩阵 $A$ 和 $B$ 在 $K$ 上相似，当且仅当它们在 $F$ 上相似，即存在一个可逆的 $n \times n$ 矩阵 $P$ 其元素来自 $K$ 使得 $B = {P}^{-1}{AP}$ 当且仅当存在一个（通常不同）的可逆的 $n \times n$ 矩阵 $Q$ 其元素来自 $F$ 使得 $B = {Q}^{-1}{AQ}$ 。

Proof: (1) Let $M$ be the rational canonical form of $A$ when computed over the smaller field $F$ . Since $M$ satisfies the conditions in the definition of the rational canonical form over $K$ ,the uniqueness of the rational canonical form implies that $M$ is also the rational canonical form of $A$ over $K$ . Hence the invariant factors of $A$ are the same whether $A$ is viewed over $F$ or over $K$ . In particular,since the minimal polynomial is the largest invariant factor of $A$ it also does not depend on the field over which $A$ is viewed. It is clear from the determinant definition of the characteristic polynomial of $A$ that this polynomial depends only on the entries of $A$ (we shall see shortly that the characteristic polynomial is the product of all the invariant factors for $A$ ,which will give an alternate proof of this result).

证明：(1) 设 $M$ 是当在较小的场 $F$ 上计算时 $A$ 的有理标准形。由于 $M$ 满足在有理标准形的定义中关于 $K$ 的条件，有理标准形的唯一性意味着 $M$ 也是 $A$ 在 $K$ 上的有理标准形。因此，$A$ 的不变因子无论是在 $F$ 上还是在 $K$ 上都是相同的。特别是，由于最小多项式是 $A$ 的最大不变因子，它也不依赖于 $A$ 所在的场。从 $A$ 的特征多项式的行列式定义中可以清楚地看出，这个多项式只依赖于 $A$ 的元素（我们很快将看到特征多项式是所有不变因子的乘积，这将为这个结果提供一个替代的证明）。

(2) If $A$ and $B$ are similar over the smaller field $F$ they are clearly similar over $K$ . Conversely,if $A$ and $B$ are similar over $K$ ,they have the same rational canonical form over $K$ . By (1) they have the same rational canonical form over $F$ ,hence are similar over $F$ by Theorem 17.

(2) 如果 $A$ 和 $B$ 在较小的场 $F$ 上相似，那么它们显然在 $K$ 上也相似。反之，如果 $A$ 和 $B$ 在 $K$ 上相似，那么它们在 $K$ 上具有相同的有理标准形。由 (1) 可知，它们在 $F$ 上也具有相同的有理标准形，因此根据定理 17，它们在 $F$ 上是相似的。

This corollary asserts in particular that the rational canonical form for an $n \times n$ matrix $A$ is an $n \times n$ matrix with entries in the smallest field containing the entries of $A$ . Further,this canonical form is the same matrix even if we allow conjugation of $A$ by nonsingular matrices whose entries come from larger fields. This explains the terminology of rational canonical form.

这个推论特别指出，对于 $n \times n$ 矩阵 $A$ 的有理标准形是一个元素在包含 $A$ 元素的最小场中的 $n \times n$ 矩阵。进一步，即使我们允许通过来自较大场的非奇异矩阵对 $A$ 进行共轭，这个标准形仍然是相同的矩阵。这就解释了为什么使用有理标准形这个术语。

The next proposition gives the connection between the characteristic polynomial of a matrix (or of a linear transformation) and its invariant factors and is quite useful for determining these invariant factors (particularly for matrices of small size).

下一个命题给出了矩阵（或线性变换）的特征多项式与其不变因子之间的联系，这对于确定这些不变因子（特别是对于小尺寸矩阵）非常有用。

Lemma 19. Let $a\left( x\right) \in F\left\lbrack x\right\rbrack$ be any monic polynomial.

引理19. 设 $a\left( x\right) \in F\left\lbrack x\right\rbrack$ 为任意首一多项式。

(1) The characteristic polynomial of the companion matrix of $a\left( x\right)$ is $a\left( x\right)$ .

(1) $a\left( x\right)$ 的伴随矩阵的特征多项式是 $a\left( x\right)$。

(2) If $M$ is the block diagonal matrix

(2) 如果 $M$ 是分块对角矩阵

given by the direct sum of matrices ${A}_{1},{A}_{2},\ldots ,{A}_{k}$ then the characteristic polynomial of $M$ is the product of the characteristic polynomials of ${A}_{1},{A}_{2},\ldots ,{A}_{k}$ .

由矩阵 ${A}_{1},{A}_{2},\ldots ,{A}_{k}$ 的直接和给定，那么 $M$ 的特征多项式是 ${A}_{1},{A}_{2},\ldots ,{A}_{k}$ 的特征多项式的乘积。

Proof: These are both straightforward exercises.

证明：这两个都是直接的练习。

Proposition 20. Let $A$ be an $n \times n$ matrix over the field $F$ .

命题20. 设 $A$ 为定义在域 $F$ 上的 $n \times n$ 矩阵。

(1) The characteristic polynomial of $A$ is the product of all the invariant factors of A.

(1) $A$ 的特征多项式是 A 的所有不变因子的乘积。

(2) (The Cayley-Hamilton Theorem) The minimal polynomial of $A$ divides the characteristic polynomial of $A$ .

(2) （凯莱-哈密顿定理）$A$ 的最小多项式整除 $A$ 的特征多项式。

(3) The characteristic polynomial of $A$ divides some power of the minimal polynomial of $A$ . In particular these polynomials have the same roots,not counting multiplicities.

(3) $A$ 的特征多项式整除 $A$ 的最小多项式的某个幂。特别地，这些多项式具有相同的根，不计重数。

The same statements are true if the matrix $A$ is replaced by a linear transformation $T$ of an $n$ -dimensional vector space over $F$ .

如果将矩阵 $A$ 替换为 $F$ 上 $n$ 维向量空间的线性变换 $T$，上述结论同样成立。

Proof: Let $B$ be the rational canonical form of $A$ . By the previous lemma the block diagonal form of $B$ shows that the characteristic polynomial of $B$ is the product of the characteristic polynomials of the companion matrices of the invariant factors of $A$ . By the first part of the lemma above, the characteristic polynomial of the companion matrix ${\mathcal{C}}_{a\left( x\right) }$ for $a\left( x\right)$ is just $a\left( x\right)$ ,which implies that the characteristic polynomial for $B$ is the product of the invariant factors of $A$ . Since $A$ and $B$ are similar,they have the same characteristic polynomial, which proves (1). Assertion (2) is immediate from (1) since the minimal polynomial for $A$ is the largest invariant factor of $A$ . The fact that all the invariant factors divide the largest one immediately implies (3). The final assertion is clear from the dictionary between linear transformations of vector spaces and matrices.

证明：设 $B$ 为 $A$ 的有理标准形。根据之前的引理，$B$ 的块对角形式表明 $B$ 的特征多项式是 $A$ 的不变因子的伴随矩阵的特征多项式的乘积。由上述引理的第一部分，$a\left( x\right)$ 的伴随矩阵 ${\mathcal{C}}_{a\left( x\right) }$ 的特征多项式就是 $a\left( x\right)$，这意味着 $B$ 的特征多项式是 $A$ 的不变因子的乘积。由于 $A$ 和 $B$ 相似，它们具有相同的特征多项式，这证明了 (1)。从 (1) 直接得出断言 (2)，因为 $A$ 的最小多项式是 $A$ 的最大不变因子。所有不变因子都能整除最大不变因子这一事实立即意味着 (3)。最后的断言从线性变换的向量空间与矩阵之间的字典关系可以清楚地看出。

Note that part (2) of the proposition is the assertion that the matrix $A$ satisfies its own characteristic polynomial,i.e., ${c}_{A}\left( A\right) = 0$ as matrices,which is the usual formulation for the Cayley-Hamilton Theorem. Note also that it implies the degree of the minimal polynomial for $A$ has degree at most $n$ ,a result mentioned before.

注意，命题的第 (2) 部分是断言矩阵 $A$ 满足其自身的特征多项式，即 ${c}_{A}\left( A\right) = 0$ 作为矩阵，这是凯莱-哈密顿定理的通常表述。还应注意，它暗示了 $A$ 的最小多项式的次数至多为 $n$，这是一个之前提到过的结果。

The relations in Proposition 20 are frequently quite useful in the determination of the invariant factors for a matrix $A$ ,particularly for matrices of small degree (cf. Exercises 3 and 4 and the examples). The following result (which relies on Exercises 16 to 19 in the previous section and whose proof we outline in the exercises) computes the invariant factors in general.

命题 20 中的关系在确定矩阵 $A$ 的不变因子时经常非常有用，特别是对于小度数的矩阵（参见练习 3 和 4 以及示例）。以下结果（依赖于上一节的练习 16 到 19，并且其证明我们在练习中概述）计算了一般情况下的不变因子。

Let $A$ be an $n \times n$ matrix over the field $F$ . Then ${xI} - A$ is an $n \times n$ matrix with entries in $F\left\lbrack x\right\rbrack$ . The three operations

设 $A$ 是定义在域 $F$ 上的一个 $n \times n$ 矩阵。那么 ${xI} - A$ 是一个其元素属于 $F\left\lbrack x\right\rbrack$ 的 $n \times n$ 矩阵。以下三种操作：

(a) interchanging two rows or columns

(a) 交换两行或两列

(b) adding a multiple (in $F\left\lbrack x\right\rbrack$ ) of one row or column to another

(b) 将一行或一列的倍数（在 $F\left\lbrack x\right\rbrack$ 中）加到另一行或另一列上

(c) multiplying any row or column by a unit in $F\left\lbrack x\right\rbrack$ ,i.e.,by a nonzero element in $F$ , are called elementary row and column operations.

(c) 将任何一行或一列乘以 $F\left\lbrack x\right\rbrack$ 中的单位元，即乘以 $F$ 中的非零元素，这些操作被称为基本的行和列变换。

Theorem 21. Let $A$ be an $n \times n$ matrix over the field $F$ . Using the three elementary row and column operations above,the $n \times n$ matrix ${xI} - A$ with entries from $F\left\lbrack x\right\rbrack$ can be put into the diagonal form (called the Smith Normal Form for $A$ )

定理21。设 $A$ 是定义在域 $F$ 上的一个 $n \times n$ 矩阵。使用上述三种基本的行和列变换，可以将具有来自 $F\left\lbrack x\right\rbrack$ 的元素的 $n \times n$ 矩阵 ${xI} - A$ 转化为对角形式（称为 $A$ 的 Smith 标准形）

with monic nonzero elements ${a}_{1}\left( x\right) ,{a}_{2}\left( x\right) ,\ldots ,{a}_{m}\left( x\right)$ of $F\left\lbrack x\right\rbrack$ with degrees at least one and satisfying ${a}_{1}\left( x\right) \left| {{a}_{2}\left( x\right) }\right| \cdots \mid {a}_{m}\left( x\right)$ . The elements ${a}_{1}\left( x\right) ,\ldots ,{a}_{m}\left( x\right)$ are the invariant factors of $A$ .

其中 ${a}_{1}\left( x\right) ,{a}_{2}\left( x\right) ,\ldots ,{a}_{m}\left( x\right)$ 是 $F\left\lbrack x\right\rbrack$ 的首一非零元素，其次数至少为1，并满足 ${a}_{1}\left( x\right) \left| {{a}_{2}\left( x\right) }\right| \cdots \mid {a}_{m}\left( x\right)$ 。元素 ${a}_{1}\left( x\right) ,\ldots ,{a}_{m}\left( x\right)$ 是 $A$ 的不变因子。

Proof: cf. the exercises.

证明：参见练习。

Invariant Factor Decomposition Algorithm: Converting to Rational Canonical Form

不变因子分解算法：转换为有理标准形

As mentioned in the exercises near the end of the previous section, keeping track of the operations necessary to diagonalize ${xI} - A$ will explicitly give a matrix $P$ such that ${P}^{-1}{AP}$ is in rational canonical form. Equivalently,if $V$ is a given $F\left\lbrack x\right\rbrack$ -module with vector space basis $\left\lbrack {{e}_{1},{e}_{2},\ldots ,{e}_{n}}\right\rbrack$ ,then $P$ defines the change of basis giving the Invariant Factor Decomposition of $V$ into a direct sum of cyclic $F\left\lbrack x\right\rbrack$ -modules. In particular,if $A$ is the matrix of the linear transformation $T$ of the $F\left\lbrack x\right\rbrack$ -module $V$ defined by $x$ (i.e., $T\left( {e}_{j}\right) = x{e}_{j} = \mathop{\sum }\limits_{{i = 1}}^{n}{a}_{ij}{e}_{i}$ where $\left. {A = \left( {a}_{ij}\right) }\right)$ ,then the matrix $P$ defines the change of basis for $V$ with respect to which the matrix for $T$ is in rational canonical form.

如前一部分末尾的练习中所述，跟踪对角化 ${xI} - A$ 所需的操作将明确给出一个矩阵 $P$，使得 ${P}^{-1}{AP}$ 处于有理标准形。等价地，如果 $V$ 是一个给定的 $F\left\lbrack x\right\rbrack$ -模，其向量空间基为 $\left\lbrack {{e}_{1},{e}_{2},\ldots ,{e}_{n}}\right\rbrack$，那么 $P$ 定义了基变换，给出 $V$ 的不变因子分解，将其分解为循环 $F\left\lbrack x\right\rbrack$ -模的直接和。特别是，如果 $A$ 是由 $x$ 定义的 $F\left\lbrack x\right\rbrack$ -模 $V$ 的线性变换 $T$ 的矩阵（即 $T\left( {e}_{j}\right) = x{e}_{j} = \mathop{\sum }\limits_{{i = 1}}^{n}{a}_{ij}{e}_{i}$ 其中 $\left. {A = \left( {a}_{ij}\right) }\right)$），那么矩阵 $P$ 定义了相对于 $V$ 的基变换，该变换使得 $T$ 的矩阵处于有理标准形。

We first describe the algorithm in the general context of determining the Invariant Factor Decomposition of a given $F\left\lbrack x\right\rbrack$ -module $V$ with vector space basis $\left\lbrack {{e}_{1},{e}_{2},\ldots ,{e}_{n}}\right\rbrack$ (the proof is outlined in the exercises). We then describe the algorithm to convert a given $n \times n$ matrix $A$ to rational canonical form (in which reference to an underlying vector space and associated linear transformation are suppressed).

我们首先在确定给定 $F\left\lbrack x\right\rbrack$ -模 $V$ 的不变因子分解的一般背景下描述算法（证明大纲在练习中给出）。然后我们描述将给定的 $n \times n$ 矩阵 $A$ 转换为有理标准形的算法（其中省略了基础向量空间和关联线性变换的引用）。

Explicit numerical examples of this algorithm are given in Examples 2 and 3 following.

本算法的具体数值示例在接下来的示例 2 和 3 中给出。

Invariant Factor Decomposition Algorithm

不变因子分解算法

Let $V$ be an $F\left\lbrack x\right\rbrack$ -module with vector space basis $\left\lbrack {{e}_{1},{e}_{2},\ldots ,{e}_{n}}\right\rbrack$ (so in particular these elements are generators for $V$ as an $F\left\lbrack x\right\rbrack$ -module). Let $T$ be the linear transformation of $V$ to itself defined by $x$ and let $A$ be the $n \times n$ matrix associated to $T$ and this choice of basis for $V$ ,i.e.,

设 $V$ 是一个 $F\left\lbrack x\right\rbrack$ -模，其向量空间基为 $\left\lbrack {{e}_{1},{e}_{2},\ldots ,{e}_{n}}\right\rbrack$（特别地，这些元素作为 $F\left\lbrack x\right\rbrack$ -模的生成元）。设 $T$ 是从 $V$ 到其自身的线性变换，定义为 $x$，设 $A$ 是与 $T$ 相关的 $n \times n$ 矩阵，以及 $V$ 的基的选择，即，

(1) Use the following three elementary row and column operations to diagonalize the matrix ${xI} - A$ over $F\left\lbrack x\right\rbrack$ ,keeping track of the row operations used:

（1）使用以下三个基本的行和列变换来对矩阵 ${xI} - A$ 进行对角化，保持对行变换的使用进行跟踪：

(a) interchange two rows or columns (which will be denoted by ${R}_{i} \leftrightarrow {R}_{j}$ for the interchange of the ${i}^{\text{th }}$ and ${j}^{\text{th }}$ rows and similarly by ${C}_{i} \leftrightarrow {C}_{j}$ for columns),

（a）交换两行或两列（交换第 ${i}^{\text{th }}$ 行和第 ${j}^{\text{th }}$ 行将用 ${R}_{i} \leftrightarrow {R}_{j}$ 表示，类似地，交换列用 ${C}_{i} \leftrightarrow {C}_{j}$ 表示），

(b) add a multiple (in $F\left\lbrack x\right\rbrack$ ) of one row or column to another (which will be denoted by ${R}_{i} + p\left( x\right) {R}_{j} \mapsto {R}_{i}$ if $p\left( x\right)$ times the ${j}^{\text{th }}$ row is added to the ${i}^{\text{th }}$ row,and similarly by ${C}_{i} + p\left( x\right) {C}_{j} \mapsto {C}_{i}$ for columns),

（b）将一行或一列的倍数（在 $F\left\lbrack x\right\rbrack$ 中）加到另一行或另一列上（如果将 $p\left( x\right)$ 倍的 ${j}^{\text{th }}$ 行加到第 ${i}^{\text{th }}$ 行上，则用 ${R}_{i} + p\left( x\right) {R}_{j} \mapsto {R}_{i}$ 表示，类似地，对于列用 ${C}_{i} + p\left( x\right) {C}_{j} \mapsto {C}_{i}$ 表示），

(c) multiply any row or column by a unit in $F\left\lbrack x\right\rbrack$ ,i.e.,by a nonzero element in $F$ (which will be denoted by $u{R}_{i}$ if the ${i}^{\text{th }}$ row is multiplied by $u \in {F}^{ \times }$ ,and similarly by $u{C}_{i}$ for columns).

（c）将任何行或列乘以 $F\left\lbrack x\right\rbrack$ 中的单位元，即 $F$ 中的非零元素（如果第 ${i}^{\text{th }}$ 行乘以 $u \in {F}^{ \times }$，则用 $u{R}_{i}$ 表示，类似地，对于列用 $u{C}_{i}$ 表示）。

(2) Beginning with the $F\left\lbrack x\right\rbrack$ -module generators $\left\lbrack {{e}_{1},{e}_{2},\ldots ,{e}_{n}}\right\rbrack$ ,for each row operation used in (1), change the set of generators by the following rules:

（2）从 $F\left\lbrack x\right\rbrack$ -模生成元 $\left\lbrack {{e}_{1},{e}_{2},\ldots ,{e}_{n}}\right\rbrack$ 开始，对于（1）中使用的每一行变换，按照以下规则改变生成元的集合：

(a) If the ${i}^{\text{th }}$ row is interchanged with the ${j}^{\text{th }}$ row then interchange the ${i}^{\text{th }}$ and ${j}^{\text{th }}$ generators.

（a）如果第 ${i}^{\text{th }}$ 行与第 ${j}^{\text{th }}$ 行交换，则交换 ${i}^{\text{th }}$ 和 ${j}^{\text{th }}$ 生成元。

(b) If $p\left( x\right)$ times the ${j}^{\text{th }}$ row is added to the ${i}^{\text{th }}$ row then subtract $p\left( x\right)$ times the ${i}^{\text{th }}$ generator from the ${j}^{\text{th }}$ generator (note the indices). (c) If the ${i}^{\text{th }}$ row is multiplied by the unit $u \in F$ then divide the ${i}^{\text{th }}$ generator by $u$ .

(b) 如果将 $p\left( x\right)$ 乘以 ${j}^{\text{th }}$ 行加到 ${i}^{\text{th }}$ 行，然后从 ${j}^{\text{th }}$ 生成器中减去 $p\left( x\right)$ 乘以 ${i}^{\text{th }}$ 生成器（注意索引）。(c) 如果 ${i}^{\text{th }}$ 行乘以单位 $u \in F$，则将 ${i}^{\text{th }}$ 生成器除以 $u$。

(3) When ${xI} - A$ has been diagonalized to the form in Theorem 21 the generators $\left\lbrack {{e}_{1},{e}_{2},\ldots ,{e}_{n}}\right\rbrack$ for $V$ will be in the form of $F\left\lbrack x\right\rbrack$ -linear combinations of ${e}_{1},{e}_{2},\ldots ,{e}_{n}$ . Use $x{e}_{j} = T\left( {e}_{j}\right) = \mathop{\sum }\limits_{{i = 1}}^{n}{a}_{ij}{e}_{i}$ to write these elements as $F$ - linear combinations of ${e}_{1},{e}_{2},\ldots ,{e}_{n}$ . When ${xI} - A$ has been diagonalized,the first $n - m$ of these linear combinations are 0 (providing a useful numerical check on the computations) and the remaining $m$ linear combinations are nonzero,i.e., the generators for $V$ are in the form $\left\lbrack {0,\ldots ,0,{f}_{1},\ldots ,{f}_{m}}\right\rbrack$ corresponding precisely to the diagonal elements in Theorem 21. The elements ${f}_{1},\ldots ,{f}_{m}$ are a set of $F\left\lbrack x\right\rbrack$ -module generators for the cyclic factors in the invariant factor decomposition of $V$ (with annihilators $\left( {{a}_{1}\left( x\right) }\right) ,\ldots ,\left( {{a}_{m}\left( x\right) }\right)$ ,respectively):

(3) 当 ${xI} - A$ 已经对角化为定理21中的形式时，$V$ 的生成器 $\left\lbrack {{e}_{1},{e}_{2},\ldots ,{e}_{n}}\right\rbrack$ 将是 $F\left\lbrack x\right\rbrack$ -线性组合的形式，由 ${e}_{1},{e}_{2},\ldots ,{e}_{n}$ 组成。使用 $x{e}_{j} = T\left( {e}_{j}\right) = \mathop{\sum }\limits_{{i = 1}}^{n}{a}_{ij}{e}_{i}$ 将这些元素写成 $F$ -线性组合的形式。当 ${xI} - A$ 已经对角化时，这些线性组合中的前 $n - m$ 个为0（为计算提供了一个有用的数值校验），其余的 $m$ 线性组合不为零，即 $V$ 的生成器是 $\left\lbrack {0,\ldots ,0,{f}_{1},\ldots ,{f}_{m}}\right\rbrack$ 的形式，精确对应定理21中的对角元素。元素 ${f}_{1},\ldots ,{f}_{m}$ 是 $F\left\lbrack x\right\rbrack$ -模生成器集合，对应于 $V$ 的不变因子分解中的循环因子（分别具有消灭子 $\left( {{a}_{1}\left( x\right) }\right) ,\ldots ,\left( {{a}_{m}\left( x\right) }\right)$）：

giving the Invariant Factor Decomposition of the $F\left\lbrack x\right\rbrack$ -module $V$ .

给出了 $F\left\lbrack x\right\rbrack$ -模 $V$ 的不变因子分解。

(4) The corresponding vector space basis for each cyclic factor of $V$ is then given by the elements ${f}_{i},T{f}_{i},{T}^{2}{f}_{i},\ldots ,{T}^{\deg {a}_{i}\left( x\right) - 1}{f}_{i}$ .

(4) 每个 $V$ 的循环因子的对应向量空间基由元素 ${f}_{i},T{f}_{i},{T}^{2}{f}_{i},\ldots ,{T}^{\deg {a}_{i}\left( x\right) - 1}{f}_{i}$ 给出。

(5) Write the ${k}^{\text{th }}$ element of the vector space basis computed in (4) in terms of the original vector space basis $\left\lbrack {{e}_{1},{e}_{2},\ldots ,{e}_{n}}\right\rbrack$ and use the coordinates for the ${k}^{\text{th}}$ column of an $n \times n$ matrix $P$ . Then ${P}^{-1}{AP}$ is in rational canonical form (with diagonal blocks the companion matrices for the ${a}_{i}\left( x\right)$ ). This is the matrix for the linear transformation $T$ with respect to the vector space basis in (4).

(5) 将在(4)中计算得到的向量空间基的${k}^{\text{th }}$元素表示为原始向量空间基$\left\lbrack {{e}_{1},{e}_{2},\ldots ,{e}_{n}}\right\rbrack$的形式，并使用${k}^{\text{th}}$列的坐标作为$n \times n$矩阵$P$的列。那么${P}^{-1}{AP}$就是有理标准形（对角块为${a}_{i}\left( x\right)$的伴随矩阵）。这是线性变换$T$相对于(4)中的向量空间基的矩阵。

We now describe the algorithm to convert a given $n \times n$ matrix $A$ to rational canonical form,i.e.,to determine an $n \times n$ matrix $P$ so that ${P}^{-1}{AP}$ is in rational canonical form. This is nothing more than the algorithm above applied to the vector space $V = {F}^{n}$ of $n \times 1$ column vectors with standard basis $\left\lbrack {{e}_{1},{e}_{2},\ldots ,{e}_{n}}\right\rbrack$ (where ${e}_{i}$ is the column vector with 1 in the ${i}^{\text{th }}$ position and 0 ’s elsewhere) and $T$ is the linear transformation defined by $A$ and this choice of basis. Explicit reference to this underlying vector space and associated linear transformation are suppressed, so the algorithm is purely matrix theoretic.

我们现在描述一个算法，将给定的$n \times n$矩阵$A$转换为有理标准形，即确定一个$n \times n$矩阵$P$，使得${P}^{-1}{AP}$为有理标准形。这不过是上述算法应用于$V = {F}^{n}$的向量空间$n \times 1$的标准基$\left\lbrack {{e}_{1},{e}_{2},\ldots ,{e}_{n}}\right\rbrack$（其中${e}_{i}$是在${i}^{\text{th }}$位置为1，其余位置为0的列向量）和由$A$以及这个基的选择定义的线性变换$T$。对这一底层向量空间和相关的线性变换的明确引用被省略，因此算法纯粹是矩阵理论上的。

Converting an $n \times n$ Matrix to Rational Canonical Form

将$n \times n$矩阵转换为有理标准形

Let $A$ be an $n \times n$ matrix with entries in the field $F$ .

设$A$是一个在字段$F$中的$n \times n$矩阵。

(1) Use the following three elementary row and column operations to diagonalize the matrix ${xI} - A$ over $F\left\lbrack x\right\rbrack$ ,keeping track of the row operations used:

(1) 使用以下三个基本的行和列操作来对矩阵${xI} - A$进行对角化，同时跟踪所使用的行操作：

(a) interchange two rows or columns (which will be denoted by ${R}_{i} \leftrightarrow {R}_{j}$ for the interchange of the ${i}^{\text{th }}$ and ${j}^{\text{th }}$ rows and similarly by ${C}_{i} \leftrightarrow {C}_{j}$ for columns),

(a) 交换两行或两列（交换第${i}^{\text{th }}$和${j}^{\text{th }}$行将表示为${R}_{i} \leftrightarrow {R}_{j}$，类似地，交换列表示为${C}_{i} \leftrightarrow {C}_{j}$），

(b) add a multiple (in $F\left\lbrack x\right\rbrack$ ) of one row or column to another (which will be denoted by ${R}_{i} + p\left( x\right) {R}_{j} \mapsto {R}_{i}$ if $p\left( x\right)$ times the ${j}^{\text{th }}$ row is added to the ${i}^{\text{th }}$ row,and similarly by ${C}_{i} + p\left( x\right) {C}_{j} \mapsto {C}_{i}$ for columns),

(b) 将一行或一列的倍数（在 $F\left\lbrack x\right\rbrack$ 中）加到另一行或列上（如果将 $p\left( x\right)$ 倍的 ${j}^{\text{th }}$ 行加到 ${i}^{\text{th }}$ 行上，则用 ${R}_{i} + p\left( x\right) {R}_{j} \mapsto {R}_{i}$ 表示，类似地，对于列则用 ${C}_{i} + p\left( x\right) {C}_{j} \mapsto {C}_{i}$ 表示），

(c) multiply any row or column by a unit in $F\left\lbrack x\right\rbrack$ ,i.e.,by a nonzero element in $F$ (which will be denoted by $u{R}_{i}$ if the ${i}^{\text{th }}$ row is multiplied by $u \in {F}^{ \times }$ ,and similarly by $u{C}_{i}$ for columns).

(c) 将任何行或列乘以 $F\left\lbrack x\right\rbrack$ 中的单位元素，即乘以 $F$ 中的非零元素（如果将 ${i}^{\text{th }}$ 行乘以 $u \in {F}^{ \times }$ ，则用 $u{R}_{i}$ 表示，类似地，对于列则用 $u{C}_{i}$ 表示）。

Define ${d}_{1},\ldots ,{d}_{m}$ to be the degrees of the monic nonconstant polynomials ${a}_{1}\left( x\right) ,\ldots ,{a}_{m}\left( x\right)$ appearing on the diagonal,respectively.

定义 ${d}_{1},\ldots ,{d}_{m}$ 为对角线上出现的首一非零多项式 ${a}_{1}\left( x\right) ,\ldots ,{a}_{m}\left( x\right)$ 的次数。

(2) Beginning with the $n \times n$ identity matrix ${P}^{\prime }$ ,for each row operation used in (1), change the matrix ${P}^{\prime }$ by the following rules:

(2) 从 $n \times n$ 单位矩阵 ${P}^{\prime }$ 开始，对于在 (1) 中使用的每一行操作，按照以下规则改变矩阵 ${P}^{\prime }$：

(a) If ${R}_{i} \leftrightarrow {R}_{j}$ then interchange the ${i}^{\text{th }}$ and ${j}^{\text{th }}$ columns of ${P}^{\prime }$ (i.e., ${C}_{i} \leftrightarrow {C}_{j}$ for $\left. {P}^{\prime }\right)$ .

(a) 如果 ${R}_{i} \leftrightarrow {R}_{j}$ ，则交换 ${P}^{\prime }$ 中的 ${i}^{\text{th }}$ 和 ${j}^{\text{th }}$ 列（即 ${C}_{i} \leftrightarrow {C}_{j}$ 对于 $\left. {P}^{\prime }\right)$）。

(b) If ${R}_{i} + p\left( x\right) {R}_{j} \mapsto {R}_{i}$ then subtract the product of the matrix $p\left( A\right)$ times the ${i}^{\text{th }}$ column of ${P}^{\prime }$ from the ${j}^{\text{th }}$ column of ${P}^{\prime }$ (i.e., ${C}_{j} - p\left( A\right) {C}_{i} \mapsto {C}_{j}$ for ${P}^{\prime }$ - note the indices).

(b) 如果 ${R}_{i} + p\left( x\right) {R}_{j} \mapsto {R}_{i}$ ，则从 ${P}^{\prime }$ 的 ${j}^{\text{th }}$ 列中减去矩阵 $p\left( A\right)$ 乘以 ${P}^{\prime }$ 的 ${i}^{\text{th }}$ 列的乘积（即 ${C}_{j} - p\left( A\right) {C}_{i} \mapsto {C}_{j}$ 对于 ${P}^{\prime }$ - 注意索引）。

(c) If $u{R}_{i}$ then divide the elements of the ${i}^{\text{th }}$ column of ${P}^{\prime }$ by $u$ (i.e., ${u}^{-1}{C}_{i}$ for $\left. {P}^{\prime }\right)$ .

(c) 如果 $u{R}_{i}$ ，则将 ${P}^{\prime }$ 的 ${i}^{\text{th }}$ 列元素除以 $u$（即 ${u}^{-1}{C}_{i}$ 对于 $\left. {P}^{\prime }\right)$）。

(3) When ${xI} - A$ has been diagonalized to the form in Theorem 21 the first $n - m$ columns of the matrix ${P}^{\prime }$ are 0 (providing a useful numerical check on the computations) and the remaining $m$ columns of ${P}^{\prime }$ are nonzero. For each $i = 1,2,\ldots ,m$ , multiply the ${i}^{\mathrm{{th}}}$ nonzero column of ${P}^{\prime }$ successively by ${A}^{0} = I,{A}^{1},{A}^{2},\ldots ,{A}^{{d}_{i} - 1},$ where ${d}_{i}$ is the integer in (1) above and use the resulting column vectors (in this order) as the next ${d}_{i}$ columns of an $n \times n$ matrix $P$ . Then ${P}^{-1}{AP}$ is in rational canonical form (whose diagonal blocks are the companion matrices for the polynomials ${a}_{1}\left( x\right) ,\ldots ,{a}_{m}\left( x\right)$ in (1)).

(3) 当 ${xI} - A$ 被对角化为定理21中的形式时，矩阵 ${P}^{\prime }$ 的前 $n - m$ 列为0（为计算提供了一个有用的数值校验），而 ${P}^{\prime }$ 的剩余 $m$ 列不为零。对于每个 $i = 1,2,\ldots ,m$ ，将 ${P}^{\prime }$ 的 ${i}^{\mathrm{{th}}}$ 不为零列依次乘以 ${A}^{0} = I,{A}^{1},{A}^{2},\ldots ,{A}^{{d}_{i} - 1},$ ，其中 ${d}_{i}$ 是上文(1)中的整数，并使用按此顺序得到的列向量作为 $n \times n$ 矩阵 $P$ 的下一个 ${d}_{i}$ 列。然后 ${P}^{-1}{AP}$ 处于有理标准形（其对角块是多项式 ${a}_{1}\left( x\right) ,\ldots ,{a}_{m}\left( x\right)$ 在(1)中的伴随矩阵）。

In the theory of canonical forms for linear transformations (or matrices) the characteristic polynomial plays the role of the order of a finite abelian group and the minimal polynomial plays the role of the exponent (after all, they are the same invariants, one for modules over the Principal Ideal Domain $\mathbb{Z}$ and the other for modules over the Principal Ideal Domain $F\left\lbrack x\right\rbrack$ ) so we can solve problems directly analogous to those we considered for finite abelian groups in Chapter 5. In particular, this includes the following:

在线性变换（或矩阵）的标准形理论中，特征多项式扮演了有限阿贝尔群阶数的作用，而最小多项式扮演了指数的作用（毕竟，它们是相同的不变量，一个适用于主理想整环 $\mathbb{Z}$ 上的模，另一个适用于主理想整环 $F\left\lbrack x\right\rbrack$ 上的模），因此我们可以直接解决与第5章中我们考虑的有限阿贝尔群类似的问题。特别是，这包括以下内容：

$\textbf{(A)}\;$ determine the rational canonical form of a given matrix (analogous to decomposing a finite abelian group as a direct product of cyclic groups)

$\textbf{(A)}\;$ 确定给定矩阵的有理标准形（类似于将有限阿贝尔群分解为循环群的直积）

(B) determine whether two given matrices are similar (analogous to determining whether two given finite abelian groups are isomorphic)

(B) 判断给定的两个矩阵是否相似（类似于判断给定的两个有限阿贝尔群是否同构）

(C) determine all similarity classes of matrices over $F$ with a given characteristic polynomial (analogous to determining all abelian groups of a given order)

(C) 确定具有给定特征多项式的 $F$ 上所有矩阵的相似类（类似于确定给定阶数的所有阿贝尔群）

(D) determine all similarity classes of $n \times n$ matrices over $F$ with a given minimal polynomial (analogous to determining all abelian groups of rank at most $n$ of a given exponent).

(D) 确定 $n \times n$ 矩阵在给定最小多项式 $F$ 上的所有相似类（类似于确定秩至多 $n$ 的给定指数的所有阿贝尔群）。

Examples

示例

(1) We find the rational canonical forms of the following matrices over $\mathbb{Q}$ and determine if they are similar:

(1) 我们找到以下 $\mathbb{Q}$ 矩阵的有理标准形，并确定它们是否相似：

A direct computation shows that all three of these matrices have the same characteristic polynomial: ${c}_{A}\left( x\right) = {c}_{B}\left( x\right) = {c}_{C}\left( x\right) = {\left( x - 2\right) }^{2}\left( {x - 3}\right)$ . Since the minimal and characteristic polynomials have the same roots, the only possibilities for the minimal polynomials are $\left( {x - 2}\right) \left( {x - 3}\right)$ or ${\left( x - 2\right) }^{2}\left( {x - 3}\right)$ . We quickly find that $\left( {A - {2I}}\right) \left( {A - {3I}}\right) = 0$ , $\left( {B - {2I}}\right) \left( {B - {3I}}\right) \neq 0$ (the 1,1-entry is nonzero) and $\left( {C - {2I}}\right) \left( {C - {3I}}\right) \neq 0$ (the 1,2-entry is nonzero). It follows that

直接计算表明这三个矩阵都有相同的特征多项式：${c}_{A}\left( x\right) = {c}_{B}\left( x\right) = {c}_{C}\left( x\right) = {\left( x - 2\right) }^{2}\left( {x - 3}\right)$ 。由于最小多项式和特征多项式有相同的根，最小多项式的唯一可能形式是 $\left( {x - 2}\right) \left( {x - 3}\right)$ 或 ${\left( x - 2\right) }^{2}\left( {x - 3}\right)$ 。我们很快发现 $\left( {A - {2I}}\right) \left( {A - {3I}}\right) = 0$ ，$\left( {B - {2I}}\right) \left( {B - {3I}}\right) \neq 0$ （1,1-项不为零）和 $\left( {C - {2I}}\right) \left( {C - {3I}}\right) \neq 0$ （1,2-项不为零）。因此

It follows immediately that there are no additional invariant factors for $B$ and $C$ . Since the invariant factors for $A$ divide the minimal polynomial and have product the characteristic polynomial,we see that $A$ has for invariant factors the polynomials $x - 2,\left( {x - 2}\right) \left( {x - 3}\right) = {x}^{2} - {5x} + 6$ . (For $2 \times 2$ and $3 \times 3$ matrices the determination of the characteristic and minimal polynomials determines all the invariant factors, cf. Exercises 3 and 4.) We conclude that $B$ and $C$ are similar and neither is similar to $A$ . The rational canonical forms are (note ${\left( x - 2\right) }^{2}\left( {x - 3}\right) = {x}^{3} - 7{x}^{2} + {16x} - {12}$ )

立刻得出 $B$ 和 $C$ 没有额外的不变因子。由于 $A$ 的不变因子除以最小多项式并乘积等于特征多项式，我们发现 $A$ 的不变因子是多项式 $x - 2,\left( {x - 2}\right) \left( {x - 3}\right) = {x}^{2} - {5x} + 6$ 。（对于 $2 \times 2$ 和 $3 \times 3$ 矩阵，特征多项式和最小多项式的确定决定了所有的不变因子，参见练习3和4。）我们得出结论，$B$ 和 $C$ 是相似的，且都不与 $A$ 相似。有理标准形是（注意 ${\left( x - 2\right) }^{2}\left( {x - 3}\right) = {x}^{3} - 7{x}^{2} + {16x} - {12}$ ）

(2) In the example above the rational canonical forms were obtained simply by determining the characteristic and minimal polynomials for the matrices. As mentioned, this is sufficient for $2 \times 2$ and $3 \times 3$ matrices since this information is sufficient to determine all of the invariant factors. For larger matrices, however, this is in general not sufficient (cf. the next example) and more work is required to determine the invariant factors. In this example we again compute the rational canonical form for the matrix $A$ in Example 1 following the two algorithms outlined above. While this is computationally more difficult for this small matrix (as will be apparent), it has the advantage even in this case that it also explicitly computes a matrix $P$ with ${P}^{-1}{AP}$ in rational canonical form.

在上述示例中，通过简单地确定矩阵的特征多项式和最小多项式来获得有理标准形。如前所述，这对于 $2 \times 2$ 和 $3 \times 3$ 矩阵来说是足够的，因为这一信息足以确定所有的不变因子。然而，对于更大的矩阵，这通常是不够的（参见下一个示例），需要更多的工作来确定不变因子。在这个示例中，我们再次计算示例 1 中矩阵 $A$ 的有理标准形，遵循上述两种算法。虽然对于这个小矩阵来说计算上更加困难（正如将出现的），但即使在每种情况下，它也有优势，即它还显式地计算了一个 $P$ 矩阵，其 ${P}^{-1}{AP}$ 在有理标准形中。

I. (Invariant Factor Decomposition) We use row and column operations (in $\mathbb{Q}\left\lbrack x\right\rbrack$ ) to reduce the matrix

I.（不变因子分解）我们使用行和列操作（在 $\mathbb{Q}\left\lbrack x\right\rbrack$ 中）来简化矩阵

to diagonal form. As in the invariant factor decomposition algorithm, we shall use the notation ${R}_{i} \leftrightarrow {R}_{j}$ to denote the interchange of the ${i}^{\text{th }}$ and ${j}^{\text{th }}$ rows, ${R}_{i} + a{R}_{j} \mapsto {R}_{i}$ if $a$ times the ${j}^{\text{th }}$ row is added to the ${i}^{\text{th }}$ row,simply $u{R}_{i}$ if the ${i}^{\text{th }}$ row is multiplied by $u$ (and similarly for columns,using $C$ instead of $R$ ). Note also that the first two operations we perform below are rather ${ad}\;{hoc}$ and were chosen simply to have integers everywhere in the computation:

到对角形式。和在不变因子分解算法中一样，我们将使用记号 ${R}_{i} \leftrightarrow {R}_{j}$ 来表示交换 ${i}^{\text{th }}$ 和 ${j}^{\text{th }}$ 行，如果 $a$ 倍的 ${j}^{\text{th }}$ 行加到 ${i}^{\text{th }}$ 行上，则是 ${R}_{i} + a{R}_{j} \mapsto {R}_{i}$ ，如果 ${i}^{\text{th }}$ 行乘以 $u$ ，则是简单地 $u{R}_{i}$（对于列操作，使用 $C$ 而不是 $R$）。请注意，下面我们执行的前两个操作相当 ${ad}\;{hoc}$ ，我们选择它们仅仅是为了在计算中到处都有整数：

This determines the invariant factors $x - 2,{x}^{2} - {5x} + 6$ for this matrix,which we determined in Example 1 above. Let now $V$ be a 3-dimensional vector space over $\mathbb{Q}$ with basis ${e}_{1},{e}_{2},{e}_{3}$ and let $T$ be the corresponding linear transformation (which defines the action of $x$ on $V$ ),i.e.,

这确定了该矩阵的不变因子 $x - 2,{x}^{2} - {5x} + 6$，我们在上面的示例 1 中确定了它们。现在让 $V$ 是一个 $\mathbb{Q}$ 上的三维向量空间，其基为 ${e}_{1},{e}_{2},{e}_{3}$，并让 $T$ 是相应的线性变换（它定义了 $x$ 在 $V$ 上的作用），即，

The row operations used in the reduction above were

在上述简化中使用的行操作是

Starting with the basis $\left\lbrack {{e}_{1},{e}_{2},{e}_{3}}\right\rbrack$ for $V$ and changing it according to the rules given in the text, we obtain

Starting with the basis $\left\lbrack {{e}_{1},{e}_{2},{e}_{3}}\right\rbrack$ for $V$ and changing it according to the rules given in the text, we obtain

Using the formulas above for the action of $x$ ,we see that these last elements are the elements $\left\lbrack {0, - 7{e}_{1} + 7{e}_{2} + {e}_{3}, - {e}_{1} + {e}_{2}}\right\rbrack$ of $V$ corresponding to the elements $1,x - 2$ and ${x}^{2} - {5x} + 6$ in the diagonalized form of ${xI} - A$ ,respectively. The elements ${f}_{1} = - 7{e}_{1} + 7{e}_{2} + {e}_{3}$ and ${f}_{2} = - {e}_{1} + {e}_{2}$ are therefore $\mathbb{Q}\left\lbrack x\right\rbrack$ -module generators for the two cyclic factors of $V$ in its invariant factor decomposition as a $\mathbb{Q}\left\lbrack x\right\rbrack$ -module. The corresponding $\mathbb{Q}$ -vector space bases for these two factors are then ${f}_{1}$ and ${f}_{2},x{f}_{2} = T{f}_{2},$ i.e., $- 7{e}_{1} + 7{e}_{2} + {e}_{3}$ and $- {e}_{1} + {e}_{2},T\left( {-{e}_{1} + {e}_{2}}\right) = - 4{e}_{1} + 3{e}_{2}.$

Using the formulas above for the action of $x$ ,we see that these last elements are the elements $\left\lbrack {0, - 7{e}_{1} + 7{e}_{2} + {e}_{3}, - {e}_{1} + {e}_{2}}\right\rbrack$ of $V$ corresponding to the elements $1,x - 2$ and ${x}^{2} - {5x} + 6$ in the diagonalized form of ${xI} - A$ ,respectively. The elements ${f}_{1} = - 7{e}_{1} + 7{e}_{2} + {e}_{3}$ and ${f}_{2} = - {e}_{1} + {e}_{2}$ are therefore $\mathbb{Q}\left\lbrack x\right\rbrack$ -module generators for the two cyclic factors of $V$ in its invariant factor decomposition as a $\mathbb{Q}\left\lbrack x\right\rbrack$ -module. The corresponding $\mathbb{Q}$ -vector space bases for these two factors are then ${f}_{1}$ and ${f}_{2},x{f}_{2} = T{f}_{2},$ i.e., $- 7{e}_{1} + 7{e}_{2} + {e}_{3}$ and $- {e}_{1} + {e}_{2},T\left( {-{e}_{1} + {e}_{2}}\right) = - 4{e}_{1} + 3{e}_{2}.$

Then the matrix

Then the matrix

conjugates $A$ into its rational canonical form:

conjugates $A$ into its rational canonical form:

as one easily checks.

as one easily checks.

II. (Converting A Directly to Rational Canonical Form) We use the row operations involved in the diagonalization of ${xI} - A$ to determine the matrix ${P}^{\prime }$ of the algorithm above:

II. (Converting A Directly to Rational Canonical Form) We use the row operations involved in the diagonalization of ${xI} - A$ to determine the matrix ${P}^{\prime }$ of the algorithm above:

Here we have ${d}_{1} = 1$ and ${d}_{2} = 2$ ,corresponding to the second and third nonzero columns of ${P}^{\prime }$ ,respectively. The columns of $P$ are therefore given by

Here we have ${d}_{1} = 1$ and ${d}_{2} = 2$ ,corresponding to the second and third nonzero columns of ${P}^{\prime }$ ,respectively. The columns of $P$ are therefore given by

respectively,which again gives the matrix $P$ above.

respectively,which again gives the matrix $P$ above.

(3) For the $3 \times 3$ matrix $A$ it was not necessary to perform the lengthy calculations above merely to determine the rational canonical form (equivalently, the invariant factors),as we saw in Example 1. For $n \times n$ matrices with $n \geq 4$ ,however,the computation of the characteristic and minimal polynomials is in general not sufficient for the determination of all the invariant factors, so the more extensive calculations of the previous example may become necessary. For example, consider the matrix

(3) For the $3 \times 3$ matrix $A$ it was not necessary to perform the lengthy calculations above merely to determine the rational canonical form (equivalently, the invariant factors),as we saw in Example 1. For $n \times n$ matrices with $n \geq 4$ ,however,the computation of the characteristic and minimal polynomials is in general not sufficient for the determination of all the invariant factors, so the more extensive calculations of the previous example may become necessary. For example, consider the matrix

A short computation shows that the characteristic polynomial of $D$ is ${\left( x - 1\right) }^{4}$ . The possible minimal polynomials are then $x - 1,{\left( x - 1\right) }^{2},{\left( x - 1\right) }^{3}$ and ${\left( x - 1\right) }^{4}$ . Clearly $D - I \neq 0$ and another short computation shows that ${\left( D - I\right) }^{2} = 0$ ,so the minimal polynomial for $D$ is ${\left( x - 1\right) }^{2}$ . There are then two possible sets of invariant factors:

简单计算表明，$D$ 的特征多项式是 ${\left( x - 1\right) }^{4}$ 。可能的最小多项式是 $x - 1,{\left( x - 1\right) }^{2},{\left( x - 1\right) }^{3}$ 和 ${\left( x - 1\right) }^{4}$ 。显然 $D - I \neq 0$ ，另一次简单计算表明 ${\left( D - I\right) }^{2} = 0$ ，因此 $D$ 的最小多项式是 ${\left( x - 1\right) }^{2}$ 。因此有两个可能的不变因子集合：

To determine the invariant factors for $D$ we apply the procedure of the previous example to the $4 \times 4$ matrix

为了确定 $D$ 的不变因子，我们将前一个例子的方法应用到 $4 \times 4$ 矩阵上

The diagonal matrix obtained from this matrix by elementary row and column opera-

通过初等行和列变换从该矩阵得到的对角矩阵是矩阵

tions is the matrix

这表明 的不变因子是 （一组使 对角化的初等行和列变换是 ，

which shows that the invariant factors for $D$ are ${\left( x - 1\right) }^{2},{\left( x - 1\right) }^{2}$ (one series of elementary row and column operations which diagonalize ${xI} - D$ are ${R}_{1} \leftrightarrow {R}_{3}, - {R}_{1}$ ,

$D$ ${\left( x - 1\right) }^{2},{\left( x - 1\right) }^{2}$ 。

${R}_{2} \leftrightarrow {R}_{4}, - {R}_{2},{R}_{3} + 2{R}_{2} \mapsto {R}_{3},{R}_{4} - \left( {x + 1}\right) {R}_{2} \mapsto {R}_{4},{C}_{3} + 2{C}_{2} \mapsto {C}_{3},$ $\left. {{C}_{4} + \left( {x - 3}\right) {C}_{2} \mapsto {C}_{4}}\right)$ .

${R}_{2} \leftrightarrow {R}_{4}, - {R}_{2},{R}_{3} + 2{R}_{2} \mapsto {R}_{3},{R}_{4} - \left( {x + 1}\right) {R}_{2} \mapsto {R}_{4},{C}_{3} + 2{C}_{2} \mapsto {C}_{3},$ $\left. {{C}_{4} + \left( {x - 3}\right) {C}_{2} \mapsto {C}_{4}}\right)$ 。

I. (Invariant Factor Decomposition) If ${e}_{1},{e}_{2},{e}_{3},{e}_{4}$ is a basis for $V$ in this case,then using the row operations in this diagonalization as in the previous example we see that the generators of $V$ corresponding to the factors above are $\left( {x - 1}\right) {e}_{1} - 2{e}_{2} - {e}_{3} = 0$ , $- 2{e}_{1} + \left( {x + 1}\right) {e}_{2} - {e}_{4} = 0,{e}_{1},{e}_{2}$ . Hence a vector space basis for the two direct factors in the invariant decomposition of $V$ in this case is given by ${e}_{1},T{e}_{1}$ and ${e}_{2},T{e}_{2}$ where The corresponding matrix $P$ relating these bases is

I. （不变因子分解）如果 ${e}_{1},{e}_{2},{e}_{3},{e}_{4}$ 是这个例子中 $V$ 的一个基，那么使用这个对角化中的行变换，就像前一个例子中那样，我们看到对应于上述因子的 $V$ 的生成元是 $\left( {x - 1}\right) {e}_{1} - 2{e}_{2} - {e}_{3} = 0$ ，$- 2{e}_{1} + \left( {x + 1}\right) {e}_{2} - {e}_{4} = 0,{e}_{1},{e}_{2}$ 。因此，在这种情况下，$V$ 的不变分解中的两个直和因子的一个向量空间基由 ${e}_{1},T{e}_{1}$ 和 ${e}_{2},T{e}_{2}$ 给出，其中对应的矩阵 $P$ 关联这些基是

so that ${P}^{-1}{DP}$ is in rational canonical form:

使得 ${P}^{-1}{DP}$ 是有理标准形：

as can easily be checked.

这可以很容易地验证。

II. (Converting D Directly to Rational Canonical Form) As in Example 2 we determine the matrix ${P}^{\prime }$ of the algorithm from the row operations used in the diagonalization of ${xI} - D :$

II. （直接将有理标准形转换为 D）就像在例子 2 中那样，我们通过在 ${xI} - D :$ 的对角化中使用的行变换确定算法的矩阵 ${P}^{\prime }$

Here we have ${d}_{1} = 2$ and ${d}_{2} = 2$ ,corresponding to the third and fourth nonzero columns of ${P}^{\prime }$ . The columns of $P$ are therefore given by

在这里，我们有 ${d}_{1} = 2$ 和 ${d}_{2} = 2$ ，对应于 ${P}^{\prime }$ 的第三和第四个非零列。因此，$P$ 的列由以下给出

respectively,which again gives the matrix $P$ above.

分别，这再次给出了上面的矩阵 $P$。

(4) In this example we determine all similarity classes of matrices $A$ with entries from $\mathbb{Q}$ with characteristic polynomial $\left( {{x}^{4} - 1}\right) \left( {{x}^{2} - 1}\right)$ . First note that any matrix with a degree

(4) 在这个例子中，我们确定所有具有 $A$ 元素的特征多项式为 $\left( {{x}^{4} - 1}\right) \left( {{x}^{2} - 1}\right)$ 的矩阵 $\mathbb{Q}$ 的相似类。首先注意，任何具有六次特征多项式的矩阵必须是一个 $A$ 矩阵。多项式 $\mathbb{Q}$ 在 $\left( {{x}^{4} - 1}\right) \left( {{x}^{2} - 1}\right)$ 中分解为不可约项为 。由于 的最小多项式 与 具有相同的根，因此 能整除 。假设 是某些 的不变因子，那么 （特别是，所有不变因子都能整除 ）和 。在这些约束条件下，唯一允许的列表是

6 characteristic polynomial must be a $6 \times 6$ matrix. The polynomial $\left( {{x}^{4} - 1}\right) \left( {{x}^{2} - 1}\right)$ factors into irreducibles in $\mathbb{Q}\left\lbrack x\right\rbrack$ as ${\left( x - 1\right) }^{2}{\left( x + 1\right) }^{2}\left( {{x}^{2} + 1}\right)$ . Since the minimal polynomial ${m}_{A}\left( x\right)$ for $A$ has the same roots as ${c}_{A}\left( x\right)$ it follows that $\left( {x - 1}\right) \left( {x + 1}\right) \left( {{x}^{2} + 1}\right)$ divides ${m}_{A}\left( x\right)$ . Suppose ${a}_{1}\left( x\right) ,\ldots ,{a}_{m}\left( x\right)$ are the invariant factors of some $A$ ,so ${a}_{m}\left( x\right) = {m}_{A}\left( x\right) ,{a}_{i}\left( x\right) \mid {a}_{i + 1}\left( x\right)$ (in particular,all the invariant factors divide ${m}_{A}\left( x\right)$ ) and ${a}_{1}\left( x\right) {a}_{2}\left( x\right) \cdots {a}_{m}\left( x\right) = \left( {{x}^{4} - 1}\right) \left( {{x}^{2} - 1}\right)$ . One easily sees that the only permissible lists under these constraints are

(5) 这个多项式分解为 ${\left( x - 1\right) }^{2}{\left( x + 1\right) }^{2}\left( {{x}^{2} + 1}\right)$ 。假设 $6 \times 6$ 是任何最小多项式能整除 $\mathbb{Q}\left\lbrack x\right\rbrack$ 的 $\left( {{x}^{4} - 1}\right) \left( {{x}^{2} - 1}\right)$ 矩阵，那么 ${\left( x - 1\right) }^{2}{\left( x + 1\right) }^{2}\left( {{x}^{2} + 1}\right)$ 。对于 $6 \times 6$ 的最小多项式的唯一限制是其次数最多为3（根据凯莱-哈密顿定理）。因此，此类矩阵 ${m}_{A}\left( x\right)$ 的最小多项式的唯一可能是

(a) $\left( {x - 1}\right) \left( {x + 1}\right) ,\;\left( {x - 1}\right) \left( {x + 1}\right) \left( {{x}^{2} + 1}\right)$

(a) $\left( {x - 1}\right) \left( {x + 1}\right) ,\;\left( {x - 1}\right) \left( {x + 1}\right) \left( {{x}^{2} + 1}\right)$

(b) $x - 1,\;\left( {x - 1}\right) {\left( x + 1\right) }^{2}\left( {{x}^{2} + 1}\right)$

(b) $x - 1,\;\left( {x - 1}\right) {\left( x + 1\right) }^{2}\left( {{x}^{2} + 1}\right)$

(c) $x + 1,\;{\left( x - 1\right) }^{2}\left( {x + 1}\right) \left( {{x}^{2} + 1}\right)$

(c) $x + 1,\;{\left( x - 1\right) }^{2}\left( {x + 1}\right) \left( {{x}^{2} + 1}\right)$

(d) ${\left( x - 1\right) }^{2}{\left( x + 1\right) }^{2}\left( {{x}^{2} + 1}\right)$ .

(d) ${\left( x - 1\right) }^{2}{\left( x + 1\right) }^{2}\left( {{x}^{2} + 1}\right)$ 。

One can now easily write out the corresponding direct sums of companion matrices to obtain representatives of the 4 similarity classes. We shall see in the next section that there are still only 4 similarity classes even in ${M}_{6}\left( \mathbb{C}\right)$ .

现在可以很容易地写出相应的伴随矩阵的直接和，以获得4个相似类的代表。我们将在下一节看到，即使在 ${M}_{6}\left( \mathbb{C}\right)$ 中，仍然只有4个相似类。

(5) In this example we find all similarity classes of $3 \times 3$ matrices $A$ with entries from $\mathbb{Q}$ satisfying ${A}^{6} = I$ . For each such $A$ ,its minimal polynomial divides ${x}^{6} - 1$ and in $\mathbb{Q}\left\lbrack x\right\rbrack$ the complete factorization of this polynomial is

(5) 在这个例子中，我们找到所有满足 ${A}^{6} = I$ 的 $3 \times 3$ 矩阵 $A$ 的相似类。对于每个这样的 $A$ ，它的最小多项式能整除 ${x}^{6} - 1$ ，在 $\mathbb{Q}\left\lbrack x\right\rbrack$ 中，这个多项式的完全分解是

Conversely,if $B$ is any $3 \times 3$ matrix whose minimal polynomial divides ${x}^{6} - 1$ ,then ${B}^{6} = I$ . The only restriction on the minimal polynomial for $B$ is that its degree is at most 3 (by the Cayley-Hamilton Theorem). The only possibilities for the minimal polynomial of such a matrix $A$ are therefore

反之，如果 $B$ 是任何最小多项式能整除 ${x}^{6} - 1$ 的 $3 \times 3$ 矩阵，那么 ${B}^{6} = I$ 。对于 $B$ 的最小多项式的唯一限制是其次数最多为3（根据凯莱-哈密顿定理）。因此，此类矩阵 $A$ 的最小多项式的唯一可能是

(a) $x - 1$

(a) $x - 1$

(b) $x + 1$

(b) $x + 1$

(c) ${x}^{2} - x + 1$

(c) ${x}^{2} - x + 1$

(d) ${x}^{2} + x + 1$

(d) ${x}^{2} + x + 1$

(e) $\left( {x - 1}\right) \left( {x + 1}\right)$

(e) $\left( {x - 1}\right) \left( {x + 1}\right)$

(f) $\left( {x - 1}\right) \left( {{x}^{2} - x + 1}\right)$

(f) $\left( {x - 1}\right) \left( {{x}^{2} - x + 1}\right)$

(g) $\left( {x - 1}\right) \left( {{x}^{2} + x + 1}\right)$

(g) $\left( {x - 1}\right) \left( {{x}^{2} + x + 1}\right)$

(b) $\left( {x + 1}\right) \left( {{x}^{2} - x + 1}\right)$

(b) $\left( {x + 1}\right) \left( {{x}^{2} - x + 1}\right)$

(i) $\left( {x + 1}\right) \left( {{x}^{2} + x + 1}\right)$ .

(i) $\left( {x + 1}\right) \left( {{x}^{2} + x + 1}\right)$ .

Under the constraints of the rational canonical form these give rise to the following permissible lists of invariant factors:

在有理标准形的约束下，这些会产生以下允许的不变因子列表：

(i) $x - 1,\;x - 1,\;x - 1$

(i) $x - 1,\;x - 1,\;x - 1$

(ii) $x + 1,x + 1,x + 1$

(ii) $x + 1,x + 1,x + 1$

(iii) $x - 1,\;\left( {x - 1}\right) \left( {x + 1}\right)$

(iii) $x - 1,\;\left( {x - 1}\right) \left( {x + 1}\right)$

(iv) $x + 1,\;\left( {x - 1}\right) \left( {x + 1}\right)$

(iv) $x + 1,\;\left( {x - 1}\right) \left( {x + 1}\right)$

(v) $\left( {x - 1}\right) \left( {{x}^{2} - x + 1}\right)$

(v) $\left( {x - 1}\right) \left( {{x}^{2} - x + 1}\right)$

(vi) $\left( {x - 1}\right) \left( {{x}^{2} + x + 1}\right)$

(vi) $\left( {x - 1}\right) \left( {{x}^{2} + x + 1}\right)$

(vii) $\left( {x + 1}\right) \left( {{x}^{2} - x + 1}\right)$

(vii) $\left( {x + 1}\right) \left( {{x}^{2} - x + 1}\right)$

(viii) $\left( {x + 1}\right) \left( {{x}^{2} + x + 1}\right)$ .

(viii) $\left( {x + 1}\right) \left( {{x}^{2} + x + 1}\right)$ .

Note that it is impossible to have a suitable set of invariant factors if the minimal polynomial is ${x}^{2} + x + 1$ or ${x}^{2} - x + 1$ . One can now write out the corresponding

注意，如果最小多项式是 ${x}^{2} + x + 1$ 或 ${x}^{2} - x + 1$，则不可能有一组合适的不变因子。现在可以写出相应的

rational canonical forms; for example,(i) is $I$ ,(ii) is $- I$ ,and (iii) is

有理标准形；例如，（i）是 $I$ ，（ii）是 $- I$ ，而（iii）是

Note also that another way of phrasing this result is that any $3 \times 3$ matrix with entries from Q whose order (multiplicatively,of course) divides 6 is similar to one of these 8 matrices, so this example determines all elements of orders 1,2,3 and 6 in the group ${\mathrm{{GL}}}_{3}\left( \mathbb{Q}\right)$ (up to similarity).

还应注意，这个结果的另一种表述方式是，任何元素来自有理数域 Q 且其阶（乘法意义上）整除 6 的 $3 \times 3$ 矩阵都与这 8 个矩阵之一相似，因此这个例子确定了群 ${\mathrm{{GL}}}_{3}\left( \mathbb{Q}\right)$ 中阶数为 1、2、3 和 6 的所有元素（相似性不变）。

EXERCISES

练习

1. Prove that similar linear transformations of $V$ (or $n \times n$ matrices) have the same characteristic and the same minimal polynomial.

2. 证明相似的线性变换 $V$（或 $n \times n$ 矩阵）具有相同的特征多项式和相同的最小多项式。

3. Let $M$ be as in Lemma 19. Prove that the minimal polynomial of $M$ is the least common multiple of the minimal polynomials of ${A}_{1},\ldots ,{A}_{k}$ .

4. 设 $M$ 如引理19中所述。证明 $M$ 的最小多项式是 ${A}_{1},\ldots ,{A}_{k}$ 的最小多项式的最小公倍数。

5. Prove that two $2 \times 2$ matrices over $F$ which are not scalar matrices are similar if and only if they have the same characteristic polynomial.

6. 证明两个在 $F$ 上的非纯量矩阵 $2 \times 2$ 相似当且仅当它们具有相同的特征多项式。

7. Prove that two $3 \times 3$ matrices are similar if and only if they have the same characteristic and same minimal polynomials. Give an explicit counterexample to this assertion for $4 \times 4$ matrices.

8. 证明两个 $3 \times 3$ 矩阵相似当且仅当它们具有相同的特征多项式和相同的最小多项式。为 $4 \times 4$ 矩阵给出一个反例来反驳这一断言。

9. Prove directly from the fact that the collection of all linear transformations of an $n$ dimensional vector space $V$ over $F$ to itself form a vector space over $F$ of dimension ${n}^{2}$ that the minimal polynomial of a linear transformation $T$ has degree at most ${n}^{2}$ .

10. 直接从所有 $n$ 维向量空间 $V$ 到其自身的线性变换集合在 $F$ 上构成一个维度为 ${n}^{2}$ 的向量空间这一事实出发，证明线性变换 $T$ 的最小多项式的次数最多为 ${n}^{2}$。

11. Prove that the constant term in the characteristic polynomial of the $n \times n$ matrix $A$ is ${\left( -1\right) }^{n}\det A$ and that the coefficient of ${x}^{n - 1}$ is the negative of the sum of the diagonal entries of $A$ (the sum of the diagonal entries of $A$ is called the trace of $A$ ). Prove that det $A$ is the product of the eigenvalues of $A$ and that the trace of $A$ is the sum of the eigenvalues of $A$ .

12. 证明 $n \times n$ 矩阵 $A$ 的特征多项式中的常数项是 ${\left( -1\right) }^{n}\det A$，并且 ${x}^{n - 1}$ 的系数是 $A$ 对角线元素之和的相反数（$A$ 对角线元素之和称为 $A$ 的迹）。证明 det $A$ 是 $A$ 特征值的乘积，并且 $A$ 的迹是 $A$ 特征值之和。

13. Determine the eigenvalues of the matrix

14. 确定矩阵的特征值

8. Verify that the characteristic polynomial of the companion matrix

9. 验证伴随矩阵的特征多项式

is

是

9. Find the rational canonical forms of

10. 找出以下矩阵的有理标准形

10. Find all similarity classes of $6 \times 6$ matrices over $\mathbb{Q}$ with minimal polynomial ${\left( x + 2\right) }^{2}\left( {x - 1}\right)$ (it suffices to give all lists of invariant factors and write out some of their corresponding matrices).

11. 找出所有在 $\mathbb{Q}$ 上具有最小多项式 ${\left( x + 2\right) }^{2}\left( {x - 1}\right)$ 的 $6 \times 6$ 矩阵的相似类（给出所有不变因子的列表并写出它们对应的某些矩阵即可）。

12. Find all similarity classes of $6 \times 6$ matrices over $\mathbb{C}$ with characteristic polynomial $\left( {{x}^{4} - 1}\right) \left( {{x}^{2} - 1}\right)$ .

13. 找出所有在 $6 \times 6$ 上具有特征多项式 $\left( {{x}^{4} - 1}\right) \left( {{x}^{2} - 1}\right)$ 的 $\mathbb{C}$ 矩阵的相似类。

14. Find all similarity classes of $3 \times 3$ matrices $A$ over ${\mathbb{F}}_{2}$ satisfying ${A}^{6} = I$ (compare with the answer we computed over $\mathbb{Q}$ ). Do the same for $4 \times 4$ matrices $B$ satisfying ${B}^{20} = I$ .

15. 找出所有在 ${\mathbb{F}}_{2}$ 上满足 ${A}^{6} = I$ 的 $3 \times 3$ 矩阵的相似类（与我们在 $\mathbb{Q}$ 上计算出的答案进行比较）。对于满足 ${B}^{20} = I$ 的 $4 \times 4$ 矩阵 $B$ 也做同样的事情。

16. Prove that the number of similarity classes of $3 \times 3$ matrices over $\mathbb{Q}$ with a given characteristic polynomial in $\mathbb{Q}\left\lbrack x\right\rbrack$ is the same as the number of similarity classes over any extension field of $\mathbb{Q}$ . Give an example to show that this is not true in general for $4 \times 4$ matrices.

17. 证明在 $\mathbb{Q}$ 上具有给定特征多项式的 $3 \times 3$ 矩阵的相似类数量与任何 $\mathbb{Q}$ 的扩展域上的相似类数量相同。举一个例子说明这对于 $4 \times 4$ 矩阵来说在一般情况下并不成立。

18. Determine all possible rational canonical forms for a linear transformation with characteristic polynomial ${x}^{2}{\left( {x}^{2} + 1\right) }^{2}$ .

19. 确定具有特征多项式 ${x}^{2}{\left( {x}^{2} + 1\right) }^{2}$ 的线性变换的所有可能的有理标准形。

20. Determine up to similarity all $2 \times 2$ rational matrices (i.e., $\in {M}_{2}\left( \mathbb{Q}\right)$ ) of precise order 4 (multiplicatively,of course). Do the same if the matrix has entries from $\mathbb{C}$ .

21. 确定 $2 \times 2$ 有理矩阵（即 $\in {M}_{2}\left( \mathbb{Q}\right)$ ）的所有相似类，其精确阶数为4（当然是指乘法意义下的阶数）。如果矩阵的元素来自 $\mathbb{C}$ ，也做同样的事情。

22. Show that ${x}^{5} - 1 = \left( {x - 1}\right) \left( {{x}^{2} - {4x} + 1}\right) \left( {{x}^{2} + {5x} + 1}\right)$ in ${\mathbb{F}}_{19}\left\lbrack x\right\rbrack$ . Use this to determine up to similarity all $2 \times 2$ matrices with entries from ${\mathbb{F}}_{19}$ of (multiplicative) order 5 .

23. 证明 ${x}^{5} - 1 = \left( {x - 1}\right) \left( {{x}^{2} - {4x} + 1}\right) \left( {{x}^{2} + {5x} + 1}\right)$ 在 ${\mathbb{F}}_{19}\left\lbrack x\right\rbrack$ 中。利用这一点确定所有在 ${\mathbb{F}}_{19}$ 中具有（乘法）阶数为5的 $2 \times 2$ 矩阵的相似类。

24. Determine representatives for the conjugacy classes for $G{L}_{3}\left( {\mathbb{F}}_{2}\right)$ . [Compare your answer with Theorem 15 and Proposition 14 of Chapter 6.]

25. 确定 $G{L}_{3}\left( {\mathbb{F}}_{2}\right)$ 的共轭类的代表。将你的答案与第6章的定理15和命题14进行比较。]

26. Let $V$ be a finite dimensional vector space over $\mathbb{Q}$ and suppose $T$ is a nonsingular linear transformation of $V$ such that ${T}^{-1} = {T}^{2} + T$ . Prove that the dimension of $V$ is divisible by 3. If the dimension of $V$ is precisely 3 prove that all such transformations $T$ are similar.

27. 设 $V$ 是 $\mathbb{Q}$ 上的有限维向量空间，假设 $T$ 是 $V$ 的一个非奇异线性变换，且满足 ${T}^{-1} = {T}^{2} + T$ 。证明 $V$ 的维数能被3整除。如果 $V$ 的维数恰好为3，证明所有这样的变换 $T$ 都是相似的。

28. Let $V$ be the infinite dimensional real vector space

29. 设 $V$ 是无限维实向量空间

Define the map $T : V \rightarrow V$ by $T\left( {{a}_{0},{a}_{1},{a}_{2},\ldots }\right) = \left( {0,{a}_{0},{a}_{1},{a}_{2},\ldots }\right)$ . Prove that $T$ has no eigenvectors.

通过 $T\left( {{a}_{0},{a}_{1},{a}_{2},\ldots }\right) = \left( {0,{a}_{0},{a}_{1},{a}_{2},\ldots }\right)$ 定义映射 $T : V \rightarrow V$ 。证明 $T$ 没有特征向量。

20. Let $\ell$ be a prime and let ${\Phi }_{\ell }\left( x\right) = \frac{{x}^{\ell } - 1}{x - 1} = {x}^{\ell - 1} + {x}^{\ell - 2} + \ldots + x + 1 \in \mathbb{Z}\left\lbrack x\right\rbrack$ be the ${\ell }^{\mathrm{{th}}}$ cyclotomic polynomial,which is irreducible over $\mathbb{Q}$ (Example 4 following Corollary 9.14). This exercise determines the smallest degree of a factor of ${\Phi }_{\ell }\left( x\right)$ modulo $p$ for any prime $p$ and so in particular determines when ${\Phi }_{\ell }\left( x\right)$ is irreducible modulo $p$ . (This actually determines the complete factorization of ${\Phi }_{\ell }\left( x\right)$ modulo $p$ - cf. Exercise 8 of Section 13.6.)

21. 设 $\ell$ 为一个质数，${\Phi }_{\ell }\left( x\right) = \frac{{x}^{\ell } - 1}{x - 1} = {x}^{\ell - 1} + {x}^{\ell - 2} + \ldots + x + 1 \in \mathbb{Z}\left\lbrack x\right\rbrack$ 为 ${\ell }^{\mathrm{{th}}}$ 的原根多项式，它在 $\mathbb{Q}$ 上是不可约的（例 4，紧随推论 9.14）。这个练习确定了对于任意质数 $p$ ，${\Phi }_{\ell }\left( x\right)$ 模 $p$ 的因子的最小次数，特别地，确定了 ${\Phi }_{\ell }\left( x\right)$ 在模 $p$ 时的不可约性。 （这实际上确定了 ${\Phi }_{\ell }\left( x\right)$ 模 $p$ 的完全分解 - 参见第 13.6 节练习 8。）

(a) Show that if $p = \ell$ then ${\Phi }_{\ell }\left( x\right)$ is divisible by $x - 1$ in ${\mathbb{F}}_{\ell }\left\lbrack x\right\rbrack$ .

(a) 证明如果 $p = \ell$ ，那么在 ${\mathbb{F}}_{\ell }\left\lbrack x\right\rbrack$ 中 ${\Phi }_{\ell }\left( x\right)$ 可被 $x - 1$ 整除。

(b) Suppose $p \neq \ell$ and let $f$ denote the order of $p$ in ${\mathbb{F}}_{\ell }^{ \times }$ ,i.e., $f$ is the smallest power of $p$ with ${p}^{f} \equiv 1{\;\operatorname{mod}\;\ell }$ . Show that $m = f$ is the first value of $m$ for which the group $G{L}_{m}\left( {\mathbb{F}}_{p}\right)$ contains an element $A$ of order $\ell$ . [Use the formula for the order of this group at the end of Section 11.1.]

(b) 假设 $p \neq \ell$ 并且设 $f$ 表示 $p$ 在 ${\mathbb{F}}_{\ell }^{ \times }$ 中的阶，即 $f$ 是 $p$ 的最小幂使得 ${p}^{f} \equiv 1{\;\operatorname{mod}\;\ell }$ 。证明 $m = f$ 是使得群 $G{L}_{m}\left( {\mathbb{F}}_{p}\right)$ 包含一个阶为 $\ell$ 的元素 $A$ 的第一个 $m$ 值。 [使用第 11.1 节末尾关于该群阶的公式。]

(c) Show that ${\Phi }_{\ell }\left( x\right)$ is not divisible by any polynomial of degree smaller than $f$ in ${\mathbb{F}}_{p}\left\lbrack x\right\rbrack$ [consider the companion matrix for such a divisor and use (b)]. Let ${m}_{A}\left( x\right) \in {\mathbb{F}}_{p}\left\lbrack x\right\rbrack$ denote the minimal polynomial for the matrix $A$ in (b) and conclude that ${m}_{A}\left( x\right)$ is irreducible of degree $f$ and divides ${\Phi }_{\ell }\left( x\right)$ in ${\mathbb{F}}_{p}\left\lbrack x\right\rbrack$ .

(c) 证明 ${\Phi }_{\ell }\left( x\right)$ 不能被 ${\mathbb{F}}_{p}\left\lbrack x\right\rbrack$ 中任何次数小于 $f$ 的多项式整除（考虑此类除数的伴随矩阵并使用 (b)）。设 ${m}_{A}\left( x\right) \in {\mathbb{F}}_{p}\left\lbrack x\right\rbrack$ 表示 (b) 中矩阵 $A$ 的最小多项式，并得出 ${m}_{A}\left( x\right)$ 是次数为 $f$ 的不可约多项式，并且在 ${\mathbb{F}}_{p}\left\lbrack x\right\rbrack$ 中整除 ${\Phi }_{\ell }\left( x\right)$。

(d) In particular,prove that ${\Phi }_{\ell }\left( x\right)$ is irreducible modulo $p$ if and only if $l - 1$ is the smallest power of $p$ which is congruent to 1 modulo $\ell$ ,i.e., $p$ is a primitive root modulo $\ell$ .

(d) 特别地，证明 ${\Phi }_{\ell }\left( x\right)$ 在模 $p$ 意义下不可约当且仅当 $l - 1$ 是 $p$ 的最小幂，满足模 $\ell$ 同余于1，即 $p$ 是模 $\ell$ 的一个原根。

21. Prove that the first two elementary row and column operations described before Theorem 21 do not change the determinant of the matrix and the third elementary operation multiplies the determinant by a unit. Conclude from Theorem 21 that the characteristic polynomial of $A$ differs by a unit from the product of the invariant factors of $A$ . Since both these polynomials are monic by definition, conclude that they are equal (this gives an alternate proof of Proposition 20).

22. 证明定理21之前描述的前两个基本行和列操作不会改变矩阵的行列式，而第三个基本操作将行列式乘以一个单位元。从定理21得出结论，$A$ 的特征多项式与 $A$ 的不变因子的乘积相差一个单位元。由于这两个多项式按定义都是首一多项式，因此它们相等（这给出了命题20的另一种证明）。

The following exercises outline the proof of Theorem 21. They carry out explicitly the construction described in Exercises 16 to 19 of the previous section for the Euclidean Domain $F\left\lbrack x\right\rbrack$ . Let $V$ be an $n$ -dimensional vector space with basis ${v}_{1},{v}_{2},\ldots ,{v}_{n}$ and let $T$ be the linear transformation of $V$ defined by the matrix $A$ and this choice of basis,i.e., $T$ is the linear transformation with

以下练习概述了定理21的证明。它们具体执行了上一节练习16至19中描述的构造，针对欧几里得域 $F\left\lbrack x\right\rbrack$ 。设 $V$ 是一个 $n$ 维向量空间，其基为 ${v}_{1},{v}_{2},\ldots ,{v}_{n}$ ，并设 $T$ 是由矩阵 $A$ 和这个基定义的 $V$ 上的线性变换，即 $T$ 是具有

where $A = \left( {a}_{ij}\right)$ . Let $F{\left\lbrack x\right\rbrack }^{n}$ be the free module of rank $n$ over $F\left\lbrack x\right\rbrack$ and let ${\xi }_{1},{\xi }_{2},\ldots ,{\xi }_{n}$ denote a basis. Then we have a natural surjective $F\left\lbrack x\right\rbrack$ -module homomorphism

其中 $A = \left( {a}_{ij}\right)$ 。设 $F{\left\lbrack x\right\rbrack }^{n}$ 是秩为 $n$ 的自由模，且 $F\left\lbrack x\right\rbrack$ 上的 ${\xi }_{1},{\xi }_{2},\ldots ,{\xi }_{n}$ 表示一个基。那么我们有一个自然的满射 $F\left\lbrack x\right\rbrack$ -模同态。

defined by mapping ${\xi }_{i}$ to ${v}_{i},i = 1,2,\ldots ,n$ . As indicated in the exercises of the previous section the invariant factors for the $F\left\lbrack x\right\rbrack$ -module $V$ can be determined once we have determined a set of generators and the corresponding relations matrix for $\ker \varphi$ . Since by definition $x$ acts on $V$ by the linear transformation $T$ ,we have

通过映射 ${\xi }_{i}$ 到 ${v}_{i},i = 1,2,\ldots ,n$ 定义。如前一部分练习所示，一旦我们确定了生成集和相应的 $\ker \varphi$ 关系矩阵，就可以确定 $F\left\lbrack x\right\rbrack$ -模 $V$ 的不变因子。由于根据定义 $x$ 通过线性变换 $T$ 作用于 $V$，因此我们有

22. Show that the elements

23. 证明元素

for $j = 1,2,\ldots ,n$ are elements of the kernel of $\varphi$ .

对于 $j = 1,2,\ldots ,n$ 是 $\varphi$ 的核中的元素。

23. (a) Show that $x{\xi }_{j} = {v}_{j} + {f}_{j}$ where ${f}_{j} \in F{\xi }_{1} + \cdots + F{\xi }_{n}$ is an element in the $F$ -vector space spanned by ${\xi }_{1},\ldots ,{\xi }_{n}$ .

24. (a) 证明 $x{\xi }_{j} = {v}_{j} + {f}_{j}$ 其中 ${f}_{j} \in F{\xi }_{1} + \cdots + F{\xi }_{n}$ 是由 ${\xi }_{1},\ldots ,{\xi }_{n}$ 生成的 $F$ -向量空间中的元素。

(b) Show that

(b) 证明

24. Show that ${v}_{1},{v}_{2},\ldots ,{v}_{n}$ generate the kernel of $\varphi$ . [Use the previous result to show that any element of $\ker \varphi$ is the sum of an element in the module generated by ${\nu }_{1},{\nu }_{2},\ldots ,{\nu }_{n}$ and an element of the form ${b}_{1}{\xi }_{1} + \cdots + {b}_{n}{\xi }_{n}$ where the ${b}_{i}$ are elements of $F$ . Then show that such an element is in $\ker \varphi$ if and only if all the ${b}_{i}$ are 0 since ${v}_{1},\ldots ,{v}_{n}$ are a basis for $V$ over $F$ .]

25. 证明 ${v}_{1},{v}_{2},\ldots ,{v}_{n}$ 生成 $\varphi$ 的核。 [使用前面的结果证明 $\ker \varphi$ 的任何元素都是 ${\nu }_{1},{\nu }_{2},\ldots ,{\nu }_{n}$ 生成的模中的元素和一个形式为 ${b}_{1}{\xi }_{1} + \cdots + {b}_{n}{\xi }_{n}$ 的元素的和，其中 ${b}_{i}$ 是 $F$ 的元素。然后证明如果且仅当所有 ${b}_{i}$ 都为0时，这样的元素在 $\ker \varphi$ 中，因为 ${v}_{1},\ldots ,{v}_{n}$ 是 $V$ 在 $F$ 上的基。]

26. Show that the generators ${v}_{1},{v}_{2},\ldots ,{v}_{n}$ of $\ker \varphi$ have corresponding relations matrix

27. 证明 $\ker \varphi$ 的生成元 ${v}_{1},{v}_{2},\ldots ,{v}_{n}$ 对应的关系矩阵为

where ${A}^{t}$ is the transpose of $A$ . Conclude that Theorem 21 and the algorithm for determining the invariant factors of $A$ follows by Exercises 16 to 19 in the previous section (note that the row and column operations necessary to diagonalize this relations matrix are the column and row operations necessary to diagonalize the matrix in Theorem 21, which explains why the invariant factor algorithm keeps track of the row operations used).

其中 ${A}^{t}$ 是 $A$ 的转置。得出结论，定理21以及确定 $A$ 不变因子的算法可以通过前一部分的练习16到19得出（注意，对角化此关系矩阵所需的行和列操作是对角化定理21中矩阵所需的列和行操作，这解释了为什么不变因子算法需要跟踪使用的行操作）。