Let T be a linear operator on a finite-dimensional vector space V with characteristic polynomial

where the ${\varphi }_i(t)$’s $(1\le i\le k)$ are distinct irreducible monic polynomials and the $n_i$’s are positive integers. For $1\le i\le k$, we define the subset ${\text{K}}_{{\varphi }_i}$ of V by

We show that each ${\text{K}}_{{\varphi }_i}$ is a nonzero T-invariant subspace of V. Note that if ${\varphi }_i(t)=t-\lambda$ is of degree one, then ${\text{K}}_{{\varphi }_i}$ is the generalized eigenspace of T corresponding to the eigenvalue $\lambda$.

Having obtained suitable generalizations of the related concepts of eigenvalue and eigenspace, our next task is to describe a canonical form of a linear operator suitable to this context. The one that we study is called the rational canonical form. Since a canonical form is a description of a matrix representation of a linear operator, it can be defined by specifying the form of the ordered bases allowed for these representations.

Here the bases of interest naturally arise from the generators of certain cyclic subspaces. For this reason, the reader should recall the definition of a T-cyclic subspace generated by a vector and Theorem 5.21 (p. 314). We briefly review this concept and introduce some new notation and terminology.

Let T be a linear operator on a finite-dimensional vector space V, and let x be a nonzero vector in V. We use the notation ${\text{C}}_x$ for the T-cyclic subspace generated by x. Recall (Theorem 5.21) that if $\dim ({\text{C}}_x)=k$, then the set

is an ordered basis for ${\text{C}}_x$. To distinguish this basis from all other ordered bases for ${\text{C}}_x$, we call it the T-cyclic basis generated by x and denote it by ${\beta }_x$. Let A be the matrix representation of the restriction of T to ${\text{C}}_x$ in the ordered basis ${\beta }_x$. Recall from the proof of Theorem 5.21 that

where

Furthermore, the characteristic polynomial of A is given by

The matrix A is called the companion matrix of the monic polynomial $h(t)=a_0+a_{1}t+\cdots +a_{k-1}{t}^{k-1}+t^k$. Every monic polynomial has a companion matrix, and the characteristic polynomial of the companion matrix of a monic polynomial g(t) of degree k is equal to ${(-1)}^{k}g(t)$. (See Exercise 19 of Section 5.4.) By Theorem 7.15 (p. 512), the monic polynomial h(t) is also the minimal polynomial of A. Since A is the matrix representation of the restriction of T to ${\text{C}}_x,h(t)$ is also the minimal polynomial of this restriction. By Exercise 15 of Section 7.3, h(t) is also the T-annihilator of x.

It is the object of this section to prove that for every linear operator T on a finite-dimensional vector space V, there exists an ordered basis $\beta$ for V such that the matrix representation ${[\text{T}]}_{\beta }$ is of the form

where each $C_i$ is the companion matrix of a polynomial ${(\varphi (t))}^m$ such that $\varphi (t)$ is a monic irreducible divisor of the characteristic polynomial of T and m is a positive integer. A matrix representation of this kind is called a rational canonical form of T. We call the accompanying basis a rational canonical basis for T.

The next theorem is a simple consequence of the following lemma, which relies on the concept of T-annihilator, introduced in the Exercises of Section 7.3.

Let T be a linear operator on a finite-dimensional vector space V, let x be a nonzero vector in V, and suppose that the T-annihilator of x is of the form ${(\varphi (t))}^p$ for some irreducible monic polynomial $\varphi (t)$. Then $\varphi (t)$ divides the minimal polynomial of T, and $x\in {\text{K}}_{\varphi }$.

By Exercise 15(b) of Section 7.3, ${(\varphi (t))}^p$ divides the minimal polynomial of T. Therefore $\varphi (t)$ divides the minimal polynomial of T. Furthermore, $x\in {\text{K}}_{\varphi }$ by the definition of ${\text{K}}_{\varphi }$.

Exercise.

If $k=1$, then (a), (b), and (e) are obvious, while (c) and (d) are vacuously true. Now suppose that $k>1$.

1. (a) The proof that ${\text{K}}_{{\varphi }_i}$ is a T-invariant subspace of V is left as an exercise. Let $f_i(t)$ be the polynomial obtained from p(t) by omitting the factor ${({\varphi }_i(t))}^{m_i}$. To prove that ${\text{K}}_{{\varphi }_i}$ is nonzero, first observe that $f_i(t)$ is a proper divisor of p(t); therefore there exists a vector $z\in \text{V}$ such that $x=f_i(\text{T})(z)\ne 0$. Then $x\in {\text{K}}_{{\varphi }_i}$ because

   $${({\varphi }_i(\text{T}))}^{m_i}(x)={({\varphi }_i(\text{T}))}^{m_i}{f}_i(\text{T})(z)=p(\text{T})(z)=0.$$

2. (b) Assume the hypothesis. Then ${({\varphi }_i(\text{T}))}^q(x)=0$ for some positive integer q. Hence the T-annihilator of x divides ${({\varphi }_i(t))}^q$ by Exercise 15(b) of Section 7.3, and the result follows.

3. (c) Assume $i\ne j$. Let $x\in {\text{K}}_{{\varphi }_i}\cap {\text{K}}_{{\varphi }_j}$, and suppose that $x\ne 0$. By (b), the T-annihilator of x is a power of both ${\varphi }_i(t)$ and ${\varphi }_j(t)$. But this is impossible because ${\varphi }_i(t)$ and ${\varphi }_j(t)$ are relatively prime (see Appendix E). We conclude that $x=0$.

4. (d) Assume $i\ne j$. Since ${\text{K}}_{{\varphi }_i}$ is T-invariant, it is also ${\varphi }_j(\text{T})$-invariant. Suppose that ${\varphi }_j(\text{T})(x)=0$ for some $x\in {\text{K}}_{{\varphi }_i}$. Then $x\in {\text{K}}_{{\varphi }_i}\cap {\text{K}}_{{\varphi }_j}=\left\{0\right\}$ by (c). Therefore the restriction of ${\varphi }_j(\text{T})$ to ${\text{K}}_{{\varphi }_i}$ is one-to-one. Since V is finite-dimensional, this restriction is also onto.

5. (e) Suppose that $1\le i\le k$. Clearly, $\text{N}({({\varphi }_i(\text{T}))}^{m_i})\subseteq {\text{K}}_{{\varphi }_i}$. Let $f_i(t)$ be the polynomial defined in (a). Since $f_i(t)$ is a product of polynomials of the form ${\varphi }_j(t)$ for $j\ne i$, we have by (d) that the restriction of $f_i(\text{T})$ to ${\text{K}}_{{\varphi }_i}$ is onto. Let $x\in {\text{K}}_{{\varphi }_i}$. Then there exists $y\in {\text{K}}_{{\varphi }_i}$ such that $f_i(\text{T})(y)=x$. Therefore

   $$({({\varphi }_i(\text{T}))}^{m_i})(x)=({({\varphi }_i(\text{T}))}^{m_i})f_i(\text{T})(y)=p(\text{T})(y)=0\text{,}$$

and hence $x\in \text{N}({({\varphi }_i(\text{T}))}^{m_i})$. Thus ${\text{K}}_{{\varphi }_i}=\text{N}({({\varphi }_i(\text{T}))}^{m_i})$.

Since a rational canonical basis for an operator T is obtained from a union of T-cyclic bases, we need to know when such a union is linearly independent. The next major result, Theorem 7.19, reduces this problem to the study of T-cyclic bases within ${\text{K}}_{\varphi }$, where $\varphi (t)$ is an irreducible monic divisor of the minimal polynomial of T. We begin with the following lemma.

Let T be a linear operator on a finite-dimensional vector space V, and suppose that

is the minimal polynomial of T, where the ${\varphi }_i$’s $(1\le i\le k)$ are the distinct irreducible monic factors of p(t) and the $m_i$’s are positive integers. For $1\le i\le k$, let $v_i\in {\text{K}}_{{\varphi }_i}$ be such that

Then $v_i=0$ for all i.

The result is trivial if $k=1$, so suppose that $k>1$. Consider any i. Let $f_i(t)$ be the polynomial obtained from p(t) by omitting the factor ${({\varphi }_i(t))}^{m_i}$. As a consequence of Theorem 7.18, $f_i(\text{T})$ is one-to-one on ${\text{K}}_{{\varphi }_i}$, and $f_i(\text{T})(v_j)=0$ for $i\ne j$. Thus, applying $f_i(\text{T})$ to (2), we obtain $f_i(\text{T})(v_i)=0$, from which it follows that $v_i=0$.

If $k=1$, then (a) is vacuously true and (b) is obvious. Now suppose that $k>1$. Then (a) follows immediately from Theorem 7.18(c). Furthermore, the proof of (b) is identical to the proof of Theorem 5.5 (p. 261) with the eigenvectors replaced by the generalized eigenvectors.

In view of Theorem 7.19, we can focus on bases of individual spaces of the form ${\text{K}}_{\varphi }$, where $\varphi (t)$ is an irreducible monic divisor of the minimal polynomial of T. The next several results give us ways to construct bases for these spaces that are unions of T-cyclic bases. These results serve the dual purposes of leading to the existence theorem for the rational canonical form and of providing methods for constructing rational canonical bases.

For Theorems 7.20 and 7.21 and the latter’s corollary, we fix a linear operator T on a finite-dimensional vector space V and an irreducible monic divisor $\varphi (t)$ of the minimal polynomial of T.

Consider any linear combination of vectors in $S_2$ that sums to zero, say,

For each i, let $f_i(t)$ be the polynomial defined by

Then (3) can be rewritten as

Apply $\varphi (\text{T})$ to both sides of (4) to obtain

This last sum can be rewritten as a linear combination of the vectors in $S_1$ so that each $f_i(\text{T})(v_i)$ is a linear combination of the vectors in ${\beta }_{v_i}$. Since $S_1$ is linearly independent, it follows that

Therefore the T-annihilator of $v_i$ divides $f_i(t)$ for all i. (See Exercise 15 of Section 7.3.) By Theorem 7.18(b), $\varphi (t)$ divides the T-annihilator of $v_i$, and hence $\varphi (t)$ divides $f_i(t)$ for all i. Thus, for each i, there exists a polynomial $g_i(t)$ such that $f_i(t)=g_i(t)\varphi (t)$. So (4) becomes

Again, linear independence of $S_1$ requires that

But $f_i(\text{T})(w_i)$ is the result of grouping the terms of the linear combination in (3) that arise from the linearly independent set ${\beta }_{w_i}$. We conclude that for each i, $a_{ij}=0$ for all j. Therefore $S_2$ is linearly independent.

We now show that ${\text{K}}_{\varphi }$ has a basis consisting of a union of T-cycles.

Let W be a T-invariant subspace of ${\text{K}}_{\varphi }$, and let $\beta$ be a basis for W. Then the following statements are true.

1. (a) Suppose that $x\in \text{N}(\varphi (\text{T}))$, but $x\notin \text{W}$. Then $\beta \cup {\beta }_x$ is linearly independent.

2. (b) For some $w_1,w_2,\dots ,w_s$ in $\text{N}(\varphi (\text{T}))$, $\beta$ can be extended to the linearly independent set

   $${\beta }^{\prime }=\beta \cup {\beta }_{w_1}\cup {\beta }_{w_2}\cup \cdots \cup {\beta }_{w_s},$$

   whose span contains $\text{N}(\varphi (\text{T}))$.

(a) Let $\beta =\{v_1,v_2,\dots ,v_k\}$, and suppose that

where d is the degree of $\varphi (t)$. Then $z\in {\text{C}}_x\cap \text{W}$, and hence ${\text{C}}_z\subseteq {\text{C}}_x\cap \text{W}$. Suppose that $z\ne 0$. Then z has $\varphi (t)$ as its T-annihilator, and therefore

It follows that ${\text{C}}_x\cap \text{W}={\text{C}}_x$, and consequently $x\in \text{W}$, contrary to hypothesis. Therefore $z=0$, from which it follows that $b_j=0$ for all j. Since $\beta$ is linearly independent, it follows that $a_i=0$ for all i. Thus $\beta \cup {\beta }_x$ is linearly independent.

(b) Suppose that W does not contain $\text{N}(\varphi (\text{T}))$. Choose a vector $w_1\in \text{N}(\varphi (\text{T}))$ that is not in W. By (a), ${\beta }_1=\beta \cup {\beta }_{w_1}$ is linearly independent. Let ${\text{W}}_1=\text{span}({\beta }_1)$. If ${\text{W}}_1$ does not contain $\text{N}(\varphi (\text{T}))$, choose a vector $w_2$ in $\text{N}(\varphi (\text{T}))$, but not in ${\text{W}}_1$, so that ${\beta }_2={\beta }_1\cup {\beta }_{w_2}=\beta \cup {\beta }_{w_1}\cup {\beta }_{w_2}$ is linearly independent. Continuing this process, we eventually obtain vectors $w_1,w_2,\dots ,w_s$ in $\text{N}(\varphi (\text{T}))$ such that the union

is a linearly independent set whose span contains $\text{N}(\varphi (\text{T}))$.

The proof is by mathematical induction on m. Suppose that $m=1$. Apply (b) of the lemma to $\text{W}=\left\{0\right\}$ to obtain a linearly independent subset of V of the form ${\beta }_{v_1}\cup {\beta }_{v_2}\cup \cdots \cup {\beta }_{v_k}$, whose span contains $\text{N}(\varphi (\text{T}))$. Since $\text{V}=\text{N}(\varphi (\text{T}))$, this set is a rational canonical basis for V.

Now suppose that, for some integer $m>1$, the result is valid whenever the minimal polynomial of T is of the form ${(\varphi (t))}^k$, where $k<m$, and assume that the minimal polynomial of T is $p(t)={(\varphi (t))}^m$. Let $r=\text{rank}(\varphi (\text{T}))$. Then $\text{R}(\varphi (\text{T}))$ is a T-invariant subspace of V, and the restriction of T to this subspace has ${(\varphi (t))}^{m-1}$ as its minimal polynomial. Therefore we may apply the induction hypothesis to obtain a rational canonical basis for the restriction of T to R(T). Suppose that $v_1,v_2,\dots ,v_k$ are the generating vectors of the T-cyclic bases that constitute this rational canonical basis. For each i, choose $w_i$ in V such that $v_i=\varphi (\text{T})(w_i)$. By Theorem 7.20, the union $\beta$ of the sets ${\beta }_{w_i}$ is linearly independent. Let $\text{W}=\text{span}(\beta )$. Then W contains $\text{R}(\varphi (\text{T}))$. Apply (b) of the lemma and adjoin additional T-cyclic bases ${\beta }_{w_{k+1}},{\beta }_{w_{k+2}},\dots ,{\beta }_{w_s}$ to $\beta$, if necessary, where $w_i$ is in $\text{N}(\varphi (\text{T}))$ for $i\ge k$, to obtain a linearly independent set

whose span ${\text{W}}^{\prime }$ contains both W and $\text{N}(\varphi (\text{T}))$.

We show that ${\text{W}}^{\prime }=\text{V}$. Let U denote the restriction of $\varphi (\text{T})$ to ${\text{W}}^{\prime }$, which is $\varphi \text{(T)}$-invariant. By the way in which ${\text{W}}^{\prime }$ was obtained from $\text{R}(\varphi (\text{T}))$, it follows that $\text{R}(\text{U})=\text{R}(\varphi (\text{T}))$ and $\text{N}(\text{U})=\text{N}(\varphi (\text{T}))$. Therefore

Thus ${\text{W}}^{\prime }=\text{V}$, and ${\beta }^{\prime }$ is a rational canonical basis for T.

${\text{K}}_{\varphi }$ has a basis consisting of the union of T-cyclic bases.

Apply Theorem 7.21 to the restriction of T to ${\text{K}}_{\varphi }$.

We are now ready to study the general case.

Let T be a linear operator on a finite-dimensional vector space V, and let $p(t)={({\varphi }_1(t))}^{m_1}{({\varphi }_2(t))}^{m_2}\cdots {({\varphi }_k(t))}^{m_k}$ be the minimal polynomial of T, where the ${\varphi }_i(t)$’s are the distinct irreducible monic factors of p(t) and $m_i>0$ for all i. The proof is by mathematical induction on k. The case $k=1$ is proved in Theorem 7.21.

Suppose that the result is valid whenever the minimal polynomial contains fewer than k distinct irreducible factors for some $k>1$, and suppose that p(t) contains k distinct factors. Let U be the restriction of T to the T-invariant subspace $\text{W}=\text{R}({({\varphi }_k(\text{T}))}^{m_k})$, and let q(t) be the minimal polynomial of U. Then q(t) divides p(t) by Exercise 10 of Section 7.3. Furthermore, ${\varphi }_k(t)$ does not divide q(t). For otherwise, there would exist a nonzero vector $x\in \text{W}$ such that ${\varphi }_k(\text{U})(x)=0$ and a vector $y\in \text{V}$ such that $x={({\varphi }_k(\text{T}))}^{m_k}(y)$. It follows that ${({\varphi }_k(\text{T}))}^{m_{k+1}}(y)=0$, and hence $y\in {\text{K}}_{{\varphi }_k}$ and $x={({\varphi }_k(\text{T}))}^{m_k}(y)=0$ by Theorem 7.18(e), a contradiction. Thus q(t) contains fewer than k distinct irreducible divisors. So by the induction hypothesis, U has a rational canonical basis ${\beta }_1$ consisting of a union of U-cyclic bases (and hence T-cyclic bases) of vectors from some of the subspaces ${\text{K}}_{{\varphi }_i},1\le i\le k-1$. By the corollary to Theorem 7.21, ${\text{K}}_{{\varphi }_k}$ has a basis ${\beta }_2$ consisting of a union of T-cyclic bases. By Theorem 7.19, ${\beta }_1$ and ${\beta }_2$ are disjoint, and $\beta ={\beta }_1\cup {\beta }_2$ is linearly independent. Let s denote the number of vectors in $\beta$.Then

We conclude that $\beta$ is a basis for V. Therefore $\beta$ is a rational canonical basis, and T has a rational canonical form.

In our study of the rational canonical form, we relied on the minimal polynomial. We are now able to relate the rational canonical form to the characteristic polynomial.

(a) By Theorem 7.22, T has a rational canonical form C. By Exercise 39 of Section 5.4, the characteristic polynomial of C, and hence of T, is the product of the characteristic polynomials of the companion matrices that compose C. Therefore each irreducible monic divisor ${\varphi }_i(t)$ of f(t) divides the characteristic polynomial of at least one of the companion matrices, and hence for some integer p, ${({\varphi }_i(t))}^p$ is the T-annihilator of a nonzero vector of V. We conclude that ${({\varphi }_i(t))}^p$, and so ${\varphi }_i(t)$, divides the minimal polynomial of T. Conversely, if $\varphi (t)$ is an irreducible monic polynomial that divides the minimal polynomial of T, then $\varphi (t)$ divides the characteristic polynomial of T because the minimal polynomial divides the characteristic polynomial.

(b), (c), and (d) Let $C={[\text{T}]}_{\beta }$, which is a rational canonical form of T. Consider any i $i(1\le i\le k)$. Since f(t) is the product of the characteristic polynomials of the companion matrices that compose C, we may multiply those characteristic polynomials that arise from the T-cyclic bases in ${\beta }_i$ to obtain the factor ${({\varphi }_i(t))}^{n_i}$ of f(t). Since this polynomial has degree $n_i{d}_i$, and the union of these bases is a linearly independent subset ${\beta }_i$ of ${\text{K}}_{{\varphi }_i}$, we have

Furthermore, $n={\displaystyle \sum _{i=1}^k{d}_i{n}_i,}$ because this sum is equal to the degree of f(t).

Now let s denote the number of vectors in $\gamma$. By Theorem 7.19, $\gamma$ is linearly independent, and therefore

Hence $n=s$, and $d_i{n}_i=\dim ({\text{K}}_{{\varphi }_i})$ for all i. It follows that $\gamma$ is a basis for V and ${\beta }_i$ is a basis for ${\text{K}}_{{\varphi }_i}$ for each i.

${\alpha }_j$ is an ordered basis for ${\text{C}}_{v_j}$.

Proof. The key to this proof is Theorem 7.4 (p. 480). Since ${\alpha }_j$ is the union of cycles of generalized eigenvectors of U corresponding to $\lambda =0$, it suffices to show that the set of initial vectors of these cycles

is linearly independent. Consider any linear combination of these vectors

where not all of the coefficients are zero. Let g(t) be the polynomial defined by $g(t)=a_0+a_{1}t+\cdots +a_{d-1}{t}^{d-1}$. Then g(t) is a nonzero polynomial of degree less than d, and hence ${(\varphi (t))}^{p_j-1}g(t)$ is a nonzero polynomial with degree less than $d{p}_j$. Since ${(\varphi (t))}^{p_j}$ is the T-annihilator of $v_j$, it follows that ${(\varphi (\text{T}))}^{p_j-1}g(\text{T})(v_j)\ne 0$. Therefore the set of initial vectors is linearly independent. So by Theorem 7.4, ${\alpha }_j$ is linearly independent, and the ${\gamma }_i$’s are disjoint. Consequently, ${\alpha }_j$ consists of $d{p}_j$ linearly independent vectors in ${\text{C}}_{v_j}$, which has dimension $d{p}_j$. We conclude that ${\alpha }_j$ is a basis for ${\text{C}}_{v_j}$.

Thus we may replace ${\beta }_{v_j}$ by ${\alpha }_j$ as a basis for ${\text{C}}_{v_j}$. We do this for each j to obtain a subset $\alpha ={\alpha }_1\cup {\alpha }_2\cup \cdots \cup {\alpha }_k$ of ${\text{K}}_{\varphi }$.

$\alpha$ is a Jordan canonical basis for ${\text{K}}_{\varphi }$.

Proof. Since ${\beta }_{v_1}\cup {\beta }_{v_2}\cup \cdots \cup {\beta }_{v_k}$ is a basis for ${\text{K}}_{\varphi }$, and since $\text{span}({\alpha }_i)=\text{span}({\beta }_{v_i})={\text{C}}_{v_i}$, Exercise 9 implies that $\alpha$ is a basis for ${\text{K}}_{\varphi }$. Because $\alpha$ is a union of cycles of generalized eigenvectors of U, we conclude that $\alpha$ is a Jordan canonical basis.

We are now in a position to relate the dot diagram of T corresponding to $\varphi (t)$ to the dot diagram of U, bearing in mind that in the first case we are considering a rational canonical form and in the second case we are considering a Jordan canonical form. For convenience, we designate the first diagram $D_1$, and the second diagram $D_2$. For each j, the presence of the T-cyclic basis ${\beta }_{x_j}$ results in a column of $p_j$ dots in $D_1$. By Lemma 1, this basis is replaced by the union ${\alpha }_j$ of d cycles of generalized eigenvectors of U, each of length $p_j$, which becomes part of the Jordan canonical basis for U. In effect, ${\alpha }_j$ determines d columns each containing $p_j$ dots in $D_2$. So each column in $D_1$ determines d columns in $D_2$ of the same length, and all columns in $D_2$ are obtained in this way. Alternatively, each row in $D_2$ has d times as many dots as the corresponding row in $D_1$. Since Theorem 7.10 (p. 493) gives us the number of dots in any row of $D_2$, we may divide the appropriate expression in this theorem by d to obtain the number of dots in the corresponding row of $D_1$. Thus we have the following result.

Under the conventions described earlier, the rational canonical form of a linear operator is unique up to the arrangement of the irreducible monic divisors of the characteristic polynomial.

Since the rational canonical form of a linear operator is unique, the polynomials corresponding to the companion matrices that determine this form are also unique. These polynomials, which are powers of the irreducible monic divisors, are called the elementary divisors of the linear operator. Since a companion matrix may occur more than once in a rational canonical form, the same is true for the elementary divisors. We call the number of such occurrences the multiplicity of the elementary divisor.

Conversely, the elementary divisors and their multiplicities determine the companion matrices and, therefore, the rational canonical form of a linear operator.