# 7.4* The Rational Canonical Form

Until now we have used eigenvalues, eigenvectors, and generalized eigenvectors in our analysis of linear operators with characteristic polynomials that split. In general, characteristic polynomials need not split, and indeed, operators need not have eigenvalues! However, the unique factorization theorem for polynomials (see page 562) guarantees that the characteristic polynomial f(t) of any linear operator T on an n-dimensional vector space factors uniquely as



where the ’s  are distinct irreducible monic polynomials and the ’s are positive integers. In the case that f(t) splits, each irreducible monic polynomial factor is of the form , where  is an eigenvalue of T, and there is a one-to-one correspondence between eigenvalues of T and the irreducible monic factors of the characteristic polynomial. In general, eigenvalues need not exist, but the irreducible monic factors always exist. In this section, we establish structure theorems based on the irreducible monic factors of the characteristic polynomial instead of eigenvalues.

In this context, the following definition is the appropriate replacement for eigenspace and generalized eigenspace.

# Definition.

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



where the ’s  are distinct irreducible monic polynomials and the ’s are positive integers. For , we define the subset  of V by



We show that each  is a nonzero T-invariant subspace of V. Note that if  is of degree one, then  is the generalized eigenspace of T corresponding to the eigenvalue .

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  for the T-cyclic subspace generated by x. Recall (Theorem 5.21) that if , then the set



is an ordered basis for . To distinguish this basis from all other ordered bases for , we call it the T-cyclic basis generated by x and denote it by . Let A be the matrix representation of the restriction of T to  in the ordered basis . 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 . 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 . (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  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  for V such that the matrix representation  is of the form



where each  is the companion matrix of a polynomial  such that  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.

# Lemma.

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  for some irreducible monic polynomial . Then  divides the minimal polynomial of T, and .

# Proof.

By Exercise 15(b) of Section 7.3,  divides the minimal polynomial of T. Therefore  divides the minimal polynomial of T. Furthermore,  by the definition of .

# Theorem 7.17.

Let T be a linear operator on a finite-dimensional vector space V, and let  be an ordered basis for V. Then  is a rational canonical basis for T if and only if  is the disjoint union of T-cyclic bases , where each  lies in  for some irreducible monic divisor  of the characteristic polynomial of T.

Exercise.

# Example 1

Suppose that T is a linear operator on  and



is a rational canonical basis for T such that

is a rational canonical form of T. In this case, the submatrices , and  are the companion matrices of the polynomials , and , respectively, where



In the context of Theorem 7.17,  is the disjoint union of the T-cyclic bases; that is,



By Exercise 39 of Section 5.4, the characteristic polynomial f(t) of T is the product of the characteristic polynomials of the companion matrices:



The rational canonical form C of the operator T in Example 1 is constructed from matrices of the form , each of which is the companion matrix of some power of a monic irreducible divisor of the characteristic polynomial of T. Furthermore, each such divisor is used in this way at least once.

In the course of showing that every linear operator T on a finite dimensional vector space has a rational canonical form C, we show that the companion matrices  that constitute C are always constructed from powers of the monic irreducible divisors of the characteristic polynomial of T. A key role in our analysis is played by the subspaces , where  is an irreducible monic divisor of the minimal polynomial of T. Since the minimal polynomial of an operator divides the characteristic polynomial of the operator, every irreducible divisor of the former is also an irreducible divisor of the latter. We eventually show that the converse is also true; that is, the minimal polynomial and the characteristic polynomial have the same irreducible divisors.

We begin with a result that lists several properties of irreducible divisors of the minimal polynomial. The reader is advised to review the definition of T-annihilator and the accompanying Exercises 15 of Section 7.3.

# Theorem 7.18.

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



is the minimal polynomial of T, where the ‘s  are the distinct irreducible monic factors of p(t) and the ’s are positive integers. Then the following statements are true.

1. (a)  is a nonzero T-invariant subspace of V for each i.

2. (b) If x is a nonzero vector in some , then the T-annihilator of x is of the form  for some integer p.

3. (c) .

4. (d)  is invariant under  for , and the restriction of  to  is one-to-one and onto.

5. (e)  for each i.

# Proof.

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

1. (a) The proof that  is a T-invariant subspace of V is left as an exercise. Let  be the polynomial obtained from p(t) by omitting the factor . To prove that  is nonzero, first observe that  is a proper divisor of p(t); therefore there exists a vector  such that . Then  because


2. (b) Assume the hypothesis. Then  for some positive integer q. Hence the T-annihilator of x divides  by Exercise 15(b) of Section 7.3, and the result follows.

3. (c) Assume . Let , and suppose that . By (b), the T-annihilator of x is a power of both  and . But this is impossible because  and  are relatively prime (see Appendix E). We conclude that .

4. (d) Assume . Since  is T-invariant, it is also -invariant. Suppose that  for some . Then  by (c). Therefore the restriction of  to  is one-to-one. Since V is finite-dimensional, this restriction is also onto.

5. (e) Suppose that . Clearly, . Let  be the polynomial defined in (a). Since  is a product of polynomials of the form  for , we have by (d) that the restriction of  to  is onto. Let . Then there exists  such that . Therefore



and hence . Thus .

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 , where  is an irreducible monic divisor of the minimal polynomial of T. We begin with the following lemma.

# 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 ’s  are the distinct irreducible monic factors of p(t) and the ’s are positive integers. For , let  be such that

 (2)

Then  for all i.

# Proof.

The result is trivial if , so suppose that . Consider any i. Let  be the polynomial obtained from p(t) by omitting the factor . As a consequence of Theorem 7.18,  is one-to-one on , and  for . Thus, applying  to (2), we obtain , from which it follows that .

# Theorem 7.19.

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



is the minimal polynomial of T, where the ‘s  are the distinct irreducible monic factors of p(t) and the ’s are positive integers. For , let  be a linearly independent subset of . Then

1.  for 

2.  is linearly independent.

# Proof.

If , then (a) is vacuously true and (b) is obvious. Now suppose that . 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 , where  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  of the minimal polynomial of T.

# Theorem 7.20.

Let  be distinct vectors in  such that



is linearly independent. For each i, suppose there exists  such that . Then



is also linearly independent.

# Proof.

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

 (3)

For each i, let  be the polynomial defined by



Then (3) can be rewritten as

 (4)

Apply  to both sides of (4) to obtain



This last sum can be rewritten as a linear combination of the vectors in  so that each  is a linear combination of the vectors in . Since  is linearly independent, it follows that



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



Again, linear independence of  requires that



But  is the result of grouping the terms of the linear combination in (3) that arise from the linearly independent set . We conclude that for each i,  for all j. Therefore  is linearly independent.

We now show that  has a basis consisting of a union of T-cycles.

# Lemma.

Let W be a T-invariant subspace of , and let  be a basis for W. Then the following statements are true.

1. (a) Suppose that , but . Then  is linearly independent.

2. (b) For some  in ,  can be extended to the linearly independent set



whose span contains .

# Proof.

(a) Let , and suppose that



where d is the degree of . Then , and hence . Suppose that . Then z has  as its T-annihilator, and therefore



It follows that , and consequently , contrary to hypothesis. Therefore , from which it follows that  for all j. Since  is linearly independent, it follows that  for all i. Thus  is linearly independent.

(b) Suppose that W does not contain . Choose a vector  that is not in W. By (a),  is linearly independent. Let . If  does not contain , choose a vector  in , but not in , so that  is linearly independent. Continuing this process, we eventually obtain vectors  in  such that the union



is a linearly independent set whose span contains .

# Theorem 7.21.

If the minimal polynomial of T is of the form , then there exists a rational canonical basis for T.

# Proof.

The proof is by mathematical induction on m. Suppose that . Apply (b) of the lemma to  to obtain a linearly independent subset of V of the form , whose span contains . Since , this set is a rational canonical basis for V.

Now suppose that, for some integer , the result is valid whenever the minimal polynomial of T is of the form , where , and assume that the minimal polynomial of T is . Let . Then  is a T-invariant subspace of V, and the restriction of T to this subspace has  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  are the generating vectors of the T-cyclic bases that constitute this rational canonical basis. For each i, choose  in V such that . By Theorem 7.20, the union  of the sets  is linearly independent. Let . Then W contains . Apply (b) of the lemma and adjoin additional T-cyclic bases  to , if necessary, where  is in  for , to obtain a linearly independent set



whose span  contains both W and .

We show that . Let U denote the restriction of  to , which is -invariant. By the way in which  was obtained from , it follows that  and . Therefore



Thus , and  is a rational canonical basis for T.

# Corollary.

 has a basis consisting of the union of T-cyclic bases.

# Proof.

Apply Theorem 7.21 to the restriction of T to .

We are now ready to study the general case.

# Theorem 7.22.

Every linear operator on a finite-dimensional vector space has a rational canonical basis and, hence, a rational canonical form.

# Proof.

Let T be a linear operator on a finite-dimensional vector space V, and let  be the minimal polynomial of T, where the ’s are the distinct irreducible monic factors of p(t) and  for all i. The proof is by mathematical induction on k. The case  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 , and suppose that p(t) contains k distinct factors. Let U be the restriction of T to the T-invariant subspace , and let q(t) be the minimal polynomial of U. Then q(t) divides p(t) by Exercise 10 of Section 7.3. Furthermore,  does not divide q(t). For otherwise, there would exist a nonzero vector  such that  and a vector  such that . It follows that , and hence  and  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  consisting of a union of U-cyclic bases (and hence T-cyclic bases) of vectors from some of the subspaces . By the corollary to Theorem 7.21,  has a basis  consisting of a union of T-cyclic bases. By Theorem 7.19,  and  are disjoint, and  is linearly independent. Let s denote the number of vectors in .Then



We conclude that  is a basis for V. Therefore  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.

# Theorem 7.23.

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



where the s  are distinct irreducible monic polynomials and the ’s are positive integers. Then the following statements are true.

1. (a)  are the irreducible monic factors of the minimal polynomial.

2. (b) For each i, , where  is the degree of .

3. (c) If  is a rational canonical basis for T, then  is a basis for  for each i.

4. (d) If  is a basis for  for each i, then  is a basis for V. In particular, if each  is a disjoint union of T-cyclic bases, then  is a rational canonical basis for T.

# Proof.

(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  of f(t) divides the characteristic polynomial of at least one of the companion matrices, and hence for some integer p,  is the T-annihilator of a nonzero vector of V. We conclude that , and so , divides the minimal polynomial of T. Conversely, if  is an irreducible monic polynomial that divides the minimal polynomial of T, then  divides the characteristic polynomial of T because the minimal polynomial divides the characteristic polynomial.

(b), (c), and (d) Let , which is a rational canonical form of T. Consider any i . 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  to obtain the factor  of f(t). Since this polynomial has degree , and the union of these bases is a linearly independent subset  of , we have



Furthermore,  because this sum is equal to the degree of f(t).

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



Hence , and  for all i. It follows that  is a basis for V and  is a basis for  for each i.

# Uniqueness of the Rational Canonical Form

Having shown that a rational canonical form exists, we are now in a position to ask about the extent to which it is unique. Certainly, the rational canonical form of a linear operator T can be modified by permuting the T-cyclic bases that constitute the corresponding rational canonical basis. This has the effect of permuting the companion matrices that make up the rational canonical form. As in the case of the Jordan canonical form, we show that except for these permutations, the rational canonical form is unique, although the rational canonical bases are not.

To simplify this task, we adopt the convention of ordering every rational canonical basis so that all the T-cyclic bases associated with the same irreducible monic divisor of the characteristic polynomial are grouped together. Furthermore, within each such grouping, we arrange the T-cyclic bases in decreasing order of size. Our task is to show that, subject to this order, the rational canonical form of a linear operator is unique up to the arrangement of the irreducible monic divisors.

As in the case of the Jordan canonical form, we introduce arrays of dots from which we can reconstruct the rational canonical form. For the Jordan canonical form, we devised a dot diagram for each eigenvalue of the given operator. In the case of the rational canonical form, we define a dot diagram for each irreducible monic divisor of the characteristic polynomial of the given operator. A proof that the resulting dot diagrams are completely determined by the operator is also a proof that the rational canonical form is unique.

In what follows, T is a linear operator on a finite-dimensional vector space with rational canonical basis  is an irreducible monic divisor of the characteristic polynomial of  are the T-cyclic bases of  that are contained in ; and d is the degree of . For each j, let  be the annihilator of . This polynomial has degree ; therefore, by Exercise 15 of Section 7.3,  contains  vectors. Furthermore,  since the T-cyclic bases are arranged in decreasing order of size. We define the dot diagram of  to be the array consisting of k columns of dots with  dots in the jth column, arranged so that the jth column begins at the top and terminates after  dots. For example, if , and , then the dot diagram is

Although each column of a dot diagram corresponds to a T-cyclic basis  in , there are fewer dots in the column than there are vectors in the basis.

# Example 2

Recall the linear operator T of Example 1 with the rational canonical basis  and the rational canonical form . Since there are two irreducible monic divisors of the characteristic polynomial of T,  and , there are two dot diagrams to consider. Because  is the T-annihilator of  and  is a basis for , the dot diagram for  consists of a single dot. The other two T-cyclic bases,  and , lie in . Since  has T-annihilator  and  has T-annihilator , in the dot diagram of  we have  and . These diagrams are as follows:

In practice, we obtain the rational canonical form of a linear operator from the information provided by dot diagrams. This is illustrated in the next example.

# Example 3

Let T be a linear operator on a finite-dimensional vector space over R, and suppose that the irreducible monic divisors of the characteristic polynomial of T are



Suppose, furthermore, that the dot diagrams associated with these divisors are as follows:

Since the dot diagram for  has two columns, it contributes two companion matrices to the rational canonical form. The first column has two dots, and therefore corresponds to the  companion matrix of . The second column, with only one dot, corresponds to the  companion matrix of . These two companion matrices are given by



The dot diagram for  consists of two columns. each containing a single dot; hence this diagram contributes two copies of the  companion matrix for , namely,



The dot diagram for  consists of a single column with a single dot contributing the single  companion matrix



Therefore the rational canonical form of T is the  matrix

We return to the general problem of finding dot diagrams. As we did before, we fix a linear operator T on a finite-dimensional vector space and an irreducible monic divisor  of the characteristic polynomial of T. Let U denote the restriction of the linear operator  to . By Theorem 7.18(d),  for some positive integer q. Consequently, by Exercise 12 of Section 7.2, the characteristic polynomial of U is , where . Therefore  is the generalized eigenspace of U corresponding to , and U has a Jordan canonical form. The dot diagram associated with the Jordan canonical form of U gives us a key to understanding the dot diagram of T that is associated with . We now relate the two diagrams.

Let  be a rational canonical basis for T, and  be the T-cyclic bases of  that are contained in . Consider one of these T-cyclic bases , and suppose again that the T-annihilator of  is . Then  consists of  vectors in . For , let  be the cycle of generalized eigenvectors of U corresponding to  with end vector , where . Then



By Theorem 7.1 (p. 478),  is a linearly independent subset of . Now let



Notice that  contains  vectors.

# Lemma 1.

 is an ordered basis for .

Proof. The key to this proof is Theorem 7.4 (p. 480). Since  is the union of cycles of generalized eigenvectors of U corresponding to , 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 . Then g(t) is a nonzero polynomial of degree less than d, and hence  is a nonzero polynomial with degree less than . Since  is the T-annihilator of , it follows that . Therefore the set of initial vectors is linearly independent. So by Theorem 7.4,  is linearly independent, and the ’s are disjoint. Consequently,  consists of  linearly independent vectors in , which has dimension . We conclude that  is a basis for .

Thus we may replace  by  as a basis for . We do this for each j to obtain a subset  of .

# Lemma 2.

 is a Jordan canonical basis for .

Proof. Since  is a basis for , and since , Exercise 9 implies that  is a basis for . Because  is a union of cycles of generalized eigenvectors of U, we conclude that  is a Jordan canonical basis.

We are now in a position to relate the dot diagram of T corresponding to  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 , and the second diagram . For each j, the presence of the T-cyclic basis  results in a column of  dots in . By Lemma 1, this basis is replaced by the union  of d cycles of generalized eigenvectors of U, each of length , which becomes part of the Jordan canonical basis for U. In effect,  determines d columns each containing  dots in . So each column in  determines d columns in  of the same length, and all columns in  are obtained in this way. Alternatively, each row in  has d times as many dots as the corresponding row in . Since Theorem 7.10 (p. 493) gives us the number of dots in any row of , we may divide the appropriate expression in this theorem by d to obtain the number of dots in the corresponding row of . Thus we have the following result.

# Theorem 7.24.

Let T be a linear operator on a finite-dimensional vector space V, let  be an irreducible monic divisor of the characteristic polynomial of T of degree d, and let  denote the number of dots in the ith row of the dot diagram for  with respect to a rational canonical basis for T. Then

1. (a) ;

2. (b)  for .

Thus the dot diagrams associated with a rational canonical form of an operator are completely determined by the operator. Since the rational canonical form is completely determined by its dot diagrams, we have the following uniqueness condition.

# Corollary.

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.

# Example 4

Let



be viewed as a subset of , the space of all real-valued functions defined on R, and let . Then V is a four-dimensional subspace of , and  is an ordered basis for V. Let D be the linear operator on V defined by , the derivative of y, and let . Then



and the characteristic polynomial of D, and hence of A, is



Thus  is the only irreducible monic divisor of f(t). Since  has degree 2 and V is four-dimensional, the dot diagram for  contains only two dots. Therefore the dot diagram is determined by , the number of dots in the first row. Because ranks are preserved under matrix representations, we can use A in place of D in the formula given in Theorem 7.24. Now



and so



It follows that the second dot lies in the second row, and the dot diagram is as follows:

Hence V is a D-cyclic space generated by a single function with D-annihilator . Furthermore, its rational canonical form is given by the companion matrix of , which is



Thus  is the only elementary divisor of D, and it has multiplicity 1. For the cyclic generator, it suffices to find a function g in V for which . Since , it follows that ; therefore  can be chosen as the cyclic generator. Hence



is a rational canonical basis for D. Notice that the function h defined by  can be chosen in place of g. This shows that the rational canonical basis is not unique.

It is convenient to refer to the rational canonical form and elementary divisors of a matrix, which are defined in the obvious way.

# Definitions.

Let . The rational canonical form of A is defined to be the rational canonical form of . Likewise, for A, the elementary divisors and their multiplicities are the same as those of .

Let A be an  matrix, let C be a rational canonical form of A, and let  be the appropriate rational canonical basis for . Then , and therefore A is similar to C. In fact, if Q is the matrix whose columns are the vectors of  in the same order, then .

# Example 5

For the following real matrix A, we find the rational canonical form C of A and a matrix Q such that .



The characteristic polynomial of A is ; therefore  and  are the distinct irreducible monic divisors of f(t). By Theorem 7.23,  and . Since the degree of  is 2, the total number of dots in the dot diagram of  is , and the number of dots  in the first row is given by



Thus the dot diagram of  is

and each column contributes the companion matrix



for  to the rational canonical form C. Consequently  is an elementary divisor with multiplicity 2. Since , the dot diagram of  consists of a single dot, which contributes the  matrix (2). Hence  is an elementary divisor with multiplicity 1. Therefore the rational canonical form C is

We can infer from the dot diagram of  that if  is a rational canonical basis for , then  is the union of two cyclic bases  and , where  and  each have annihilator . It follows that both  and  lie in . It can be shown that



is a basis for . Setting , we see that



Next choose  in , but not in the span of . For example, . Then it can be seen that



and  is a basis for .

Since the dot diagram of  consists of a single dot, any nonzero vector in  is an eigenvector of A corresponding to the eigenvalue . For example, choose



By Theorem 7.23,  is a rational canonical basis for . So setting



we have .

# Example 6

For the following matrix A, we find the rational canonical form C and a matrix Q such that .



Since the characteristic polynomial of A is , the only irreducible monic divisor of f(t) is , and so . In this case,  has degree 1; hence in applying Theorem 7.24 to compute the dot diagram for , we obtain



and



where  is the number of dots in the ith row of the dot diagram. Since there are  dots in the diagram, we may terminate these computations with . Thus the dot diagram for A is

Since  has the companion matrix



and  has the companion matrix (2), the rational canonical form of A is given by



Next we find a rational canonical basis for . The preceding dot diagram indicates that there are two vectors  and  in  with annihilators  and , respectively, and such that



is a rational canonical basis for . Furthermore, , and . It can easily be shown that



and



The standard vector  meets the criteria for ; so we set . It follows that



Next we choose a vector  that is not in the span of . Clearly,  satisfies this condition. Thus



is a rational canonical basis for .

Finally, let Q be the matrix whose columns are the vectors of  in the same order:



Then .

# Direct Sums*

The next theorem is a simple consequence of Theorem 7.23.

# Theorem 7.25. (Primary Decomposition Theorem)

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



where the  are distinct irreducible monic polynomials and the ’s are positive integers. Then the following statements are true.

1. .

2. If  is the restriction of T to  and  is the rational canonical form of , then  is the rational canonical form of T.

# Proof.

Exercise.

The next theorem is a simple consequence of Theorem 7.17.

# Theorem 7.26.

Let T be a linear operator on a finite-dimensional vector space V. Then V is a direct sum of T-cyclic subspaces , where each  lies in  for some irreducible monic divisor  of the characteristic polynomial of T.

Exercise.

# Exercises

1. Label the following statements as true or false.

1. (a) Every rational canonical basis for a linear operator T is the union of T-cyclic bases.

2. (b) If a basis is the union of T-cyclic bases for a linear operator T, then it is a rational canonical basis for T.

3. (c) There exist square matrices having no rational canonical form.

4. (d) A square matrix is similar to its rational canonical form.

5. (e) For any linear operator T on a finite-dimensional vector space, any irreducible factor of the characteristic polynomial of T divides the minimal polynomial of T.

6. (f) Let  be an irreducible monic divisor of the characteristic polynomial of a linear operator T. The dots in the diagram used to compute the rational canonical form of the restriction of T to  are in one-to-one correspondence with the vectors in a basis for .

7. (g) If a matrix has a Jordan canonical form, then its Jordan canonical form and rational canonical form are similar.

2. For each of the following matrices , find the rational canonical form C of A and a matrix  such that .

1. (a) 

2. (b) 

3. (c) 

4. (d) 

5. (e) 

3. For each of the following linear operators T, find the elementary divisors, the rational canonical form C, and a rational canonical basis .

1. (a) T is the linear operator on  defined by


2. (b) Let , a subset of , and let . Define T to be the linear operator on V such that


3. (c) T is the linear operator on  defined by


4. (d) Let , a subset of , and let . Define T to be the linear operator on V such that


4. Let T be a linear operator on a finite-dimensional vector space V with minimal polynomial  for some positive integer m.

1. (a) Prove that .

2. (b) Give an example to show that the subspaces in (a) need not be equal.

3. (c) Prove that the minimal polynomial of the restriction of T to  equals .

5. Let T be a linear operator on a finite-dimensional vector space. Prove that the rational canonical form of T is a diagonal matrix if and only if T is diagonalizable. Visit goo.gl/tK8pru for a solution.

6. Let T be a linear operator on a finite-dimensional vector space V with characteristic polynomial , where  and  are distinct irreducible monic polynomials and .

1. (a) Prove that there exist  such that  has T-annihilator  has T-annihilator , and  is a basis for V.

2. (b) Prove that there is a vector  with T-annihilator  such that  is a basis for V.

3. (c) Describe the difference between the matrix representation of T with respect to  and the matrix representation of T with respect to .

Thus, to assure the uniqueness of the rational canonical form, we require that the generators of the T-cyclic bases that constitute a rational canonical basis have T-annihilators equal to powers of irreducible monic factors of the characteristic polynomial of T.

7. Let T be a linear operator on a finite-dimensional vector space with minimal polynomial



where the ’s are distinct irreducible monic factors of f(t). Prove that for each i,  is the number of entries in the first column of the dot diagram for .

8. Let T be a linear operator on a finite-dimensional vector space V. Prove that for any irreducible polynomial , if  is not one-to-one, then  divides the characteristic polynomial of T. Hint: Apply Exercise 15 of Section 7.3.

9. Let V be a vector space and  be disjoint subsets of V whose union is a basis for V. Now suppose that  are linearly independent subsets of V such that  for all i. Prove that  is also a basis for V.

10. Let T be a linear operator on a finite-dimensional vector space, and suppose that  is an irreducible monic factor of the characteristic polynomial of T. Prove that if  is the T-annihilator of vectors x and y, then  if and only if .

Exercises 11 and 12 are concerned with direct sums.

1. Prove Theorem 7.25.

2. Prove Theorem 7.26.