# 7.2 The Jordan Canonical Form II

For the purposes of this section, we fix a linear operator T on an n-dimensional vector space V such that the characteristic polynomial of T splits. Let  be the distinct eigenvalues of T.

By Theorem 7.7 (p. 484), each generalized eigenspace  contains an ordered basis  consisting of a union of disjoint cycles of generalized eigenvectors corresponding to . So by Theorems 7.4(b) (p. 480) and 7.5 (p. 482), the union  is a Jordan canonical basis for T. For each i, let  be the restriction of T to , and let . Then  is the Jordan canonical form of , and



is the Jordan canonical form of T. In this matrix, each O is a zero matrix of appropriate size.

In this section, we compute the matrices  and the bases , thereby computing J and  as well. While developing a method for finding J, it becomes evident that in some sense the matrices  are unique.

To aid in formulating the uniqueness theorem for J, we adopt the following convention: The basis  for  will henceforth be ordered in such a way that the cycles appear in order of decreasing length. That is, if  is a disjoint union of cycles  and if the length of the cycle  is , we index the cycles so that . This ordering of the cycles limits the possible orderings of vectors in , which in turn determines the matrix . It is in this sense that  is unique. It then follows that the Jordan canonical form for T is unique up to an ordering of the eigenvalues of T. As we will see, there is no uniqueness theorem for the bases  or for . However, we show that for each i, the number  of cycles that form , and the length  of each cycle, is completely determined by T.

# Example 1

To illustrate the discussion above, suppose that, for some i, the ordered basis  for  is the union of four cycles  with respective lengths , and . Then

To help us visualize each of the matrices  and ordered bases , we use an array of dots called a dot diagram of , where  is the restriction of T to . Suppose that  is a disjoint union of cycles of generalized eigenvectors  with lengths , respectively. The dot diagram of  contains one dot for each vector in , and the dots are configured according to the following rules.

1. The array consists of  columns (one column for each cycle).

2. Counting from left to right, the jth column consists of the  dots that correspond to the vectors of  starting with the initial vector at the top and continuing down to the end vector.

Denote the end vectors of the cycles by . In the following dot diagram of , each dot is labeled with the name of the vector in  to which it corresponds.

Notice that the dot diagram of  has  columns (one for each cycle) and  rows. Since , the columns of the dot diagram become shorter (or at least not longer) as we move from left to right.

Now let  denote the number of dots in the jth row of the dot diagram. Observe that . Furthermore, the diagram can be reconstructed from the values of the ’s. The proofs of these facts, which are combinatorial in nature, are treated in Exercise 9.

In Example 1, with , and , the dot diagram of  is as follows:

Here , and .

We now devise a method for computing the dot diagram of  using the ranks of linear operators determined by T and . It will follow that the dot diagram is completely determined by T, from which it will follow that it is unique. On the other hand,  is not unique. For example, see Exercise 8. (It is for this reason that we associate the dot diagram with  rather than with .)

To determine the dot diagram of , we devise a method for computing each , the number of dots in the jth row of the dot diagram, using only T and . The next three results give us the required method. To facilitate our arguments, we fix a basis  for  so that  is a disjoint union of  cycles of generalized eigenvectors with lengths .

# Theorem 7.9.

For any positive integer r, the vectors in  that are associated with the dots in the first r rows of the dot diagram of  constitute a basis for . Hence the number of dots in the first r rows of the dot diagram equals .

# Proof.

Clearly, , and  is invariant under . Let U denote the restriction of  to . By the preceding remarks, , and hence it suffices to establish the theorem for U. Now define



Let a and b denote the number of vectors in  and , respectively, and let . Then . For any  if and only if x is one of the first r vectors of a cycle, and this is true if and only if x corresponds to a dot in the first r rows of the dot diagram. Hence a is the number of dots in the first r rows of the dot diagram. For any , the effect of applying U to x is to move the dot corresponding to x exactly r places up its column to another dot. It follows that U maps  in a one-to-one fashion into . Thus  is a basis for R(U) consisting of b vectors. Hence , and so . But  is a linearly independent subset of N(U) consisting of a vectors; therefore  is a basis for N(U).

In the case that , Theorem 7.9 yields the following corollary.

# Corollary.

The dimension of  is . Hence in a Jordan canonical form of T, the number of Jordan blocks corresponding to  equals the dimension of .

# Proof.

Exercise.

We are now able to devise a method for describing the dot diagram in terms of the ranks of operators.

# Theorem 7.10.

Let  denote the number of dots in the jth row of the dot diagram of , the restriction of T to . Then the following statements are true.

1. 

2. 

if .

# Proof.

By Theorem 7.9, for , we have



Hence



and for ,



Theorem 7.10 shows that the dot diagram of  is completely determined by T and . Hence we have proved the following result.

# Corollary.

For any eigenvalue  of T, the dot diagram of  is unique. Thus, subject to the convention that the cycles of generalized eigenvectors for the bases of each generalized eigenspace are listed in order of decreasing length, the Jordan canonical form of a linear operator or a matrix is unique up to the ordering of the eigenvalues.

We apply these results to find the Jordan canonical forms of two matrices and a linear operator.

# Example 2

Let



We find the Jordan canonical form of A and a Jordan canonical basis for the linear operator . The characteristic polynomial of A is



Thus A has two distinct eigenvalues,  and , with multiplicities 3 and 1, respectively. Let  and  be the restrictions of  to the generalized eigenspaces  and , respectively.

Suppose that  is a Jordan canonical basis for . Since  has multiplicity 3, it follows that  by Theorem 7.4(c) (p. 480); hence the dot diagram of  has three dots. As we did earlier, let  denote the number of dots in the jth row of this dot diagram. Then, by Theorem 7.10,



and



(Actually, the computation of  is unnecessary in this case because  and the dot diagram only contains three dots.) Hence the dot diagram associated with  is

So



Since  has multiplicity 1, it follows that , and consequently any basis  for  consists of a single eigenvector corresponding to . Therefore



Setting , we have



and so J is the Jordan canonical form of A.

We now find a Jordan canonical basis for . We begin by determining a Jordan canonical basis  for . Since the dot diagram of  has two columns, each corresponding to a cycle of generalized eigenvectors, there are two such cycles. Let  and  denote the end vectors of the first and second cycles, respectively. We reprint below the dot diagram with the dots labeled with the names of the vectors to which they correspond.

From this diagram we see that  but . Now



It is easily seen that



is a basis for . Of these three basis vectors, the last two do not belong to , and hence we select one of these for . Suppose that we choose



Then



Now simply choose  to be a vector in  that is linearly independent of ; for example, select



Thus we have associated the Jordan canonical basis



with the dot diagram in the following manner.

By Theorem 7.6 (p. 483), the linear independence of  is guaranteed since  was chosen to be linearly independent of .

Since  has multiplicity 1, . Hence any eigenvector of  corresponding to  constitutes an appropriate basis . For example,



Thus



is a Jordan canonical basis for .

Notice that if



then .

# Example 3

Let



We find the Jordan canonical form J of A, a Jordan canonical basis for , and a matrix Q such that .

The characteristic polynomial of A is . Let , and , and let  be the restriction of  to  for .

We begin by computing the dot diagram of . Let  denote the number of dots in the first row of this diagram. Then



hence the dot diagram of  is as follows.



Therefore



where  is any basis corresponding to the dots. In this case,  is an arbitrary basis for , for example,



Next we compute the dot diagram of . Since , there is only  dot in the first row of the diagram. Since  has multiplicity 2, we have , and hence this dot diagram has the following form:



Thus



where  is any basis for  corresponding to the dots. In this case,  is a cycle of length 2. The end vector of this cycle is a vector  such that . One way of finding such a vector was used to select the vector  in Example 2. In this example, we illustrate another method. A simple calculation shows that a basis for the null space of  is



Choose v to be any solution to the system of linear equations



for example,



Thus



Therefore



is a Jordan canonical basis for . The corresponding Jordan canonical form is given by



Finally, we define Q to be the matrix whose columns are the vectors of  listed in the same order, namely,



Then .

# Example 4

Let V be the vector space of polynomial functions in two real variables x and y of degree at most 2. Then V is a vector space over R and  is an ordered basis for V. Let T be the linear operator on V defined by



For example, if , then



We find the Jordan canonical form and a Jordan canonical basis for T.

Let . Then



and hence the characteristic polynomial of T is



Thus  is the only eigenvalue of T, and . For each j, let  denote the number of dots in the jth row of the dot diagram of T. By Theorem 7.10,



and since



.

Because there are a total of six dots in the dot diagram and  and , it follows that . So the dot diagram of T is

We conclude that the Jordan canonical form of T is

We now find a Jordan canonical basis for T. Since the first column of the dot diagram of T consists of three dots, we must find a polynomial  such that . Examining the basis  for , we see that  is a suitable candidate. Setting , we see that



and



Likewise, since the second column of the dot diagram consists of two dots, we must find a polynomial  such that



Since our choice must be linearly independent of the polynomials already chosen for the first cycle, the only choice in  that satisfies these constraints is xy. So we set . Thus



Finally, the third column of the dot diagram consists of a single polynomial that lies in the null space of T. The only remaining polynomial in  is , and it is suitable here. So set . Therefore we have identified polynomials with the dots in the dot diagram as follows.

Thus  is a Jordan canonical basis for T.

In the three preceding examples, we relied on our ingenuity and the context of the problem to find Jordan canonical bases. The reader can do the same in the exercises. We are successful in these cases because the dimensions of the generalized eigenspaces under consideration are small. We do not attempt, however, to develop a general algorithm for computing Jordan canonical bases, although one could be devised by following the steps in the proof of the existence of such a basis (Theorem 7.7 p. 484).

The following result may be thought of as a corollary to Theorem 7.10.

# Theorem 7.11.

Let A and B be  matrices, each having Jordan canonical forms computed according to the conventions of this section. Then A and B are similar if and only if they have (up to an ordering of their eigenvalues) the same Jordan canonical form.

# Proof.

If A and B have the same Jordan canonical form J, then A and B are each similar to J and hence are similar to each other.

Conversely, suppose that A and B are similar. Then A and B have the same eigenvalues. Let  and  denote the Jordan canonical forms of A and B, respectively, with the same ordering of their eigenvalues. Then A is similar to both  and , and therefore, by the corollary to Theorem 2.23 (p. 115),  and  are matrix representations of . Hence  and  are Jordan canonical forms of . Thus  by the corollary to Theorem 7.10.

# Example 5

We determine which of the matrices



are similar. Observe that A, B, and C have the same characteristic polynomial , whereas D has  as its characteristic polynomial. Because similar matrices have the same characteristic polynomials, D cannot be similar to A, B, or C. Let , and  be the Jordan canonical forms of A, B, and C, respectively, using the ordering 1, 2 for their common eigenvalues. Then (see Exercise 4)



Since , A is similar to C. Since  is different from  and , B is similar to neither A nor C.

The reader should observe that any diagonal matrix is a Jordan canonical form. Thus a linear operator T on a finite-dimensional vector space V is diagonalizable if and only if its Jordan canonical form is a diagonal matrix. Hence T is diagonalizable if and only if the Jordan canonical basis for T consists of eigenvectors of T. Similar statements can be made about matrices. Thus, of the matrices A, B, and C in Example 5, A and C are not diagonalizable because their Jordan canonical forms are not diagonal matrices.

# Exercises

1. Label the following statements as true or false. Assume that the characteristic polynomial of the matrix or linear operator splits.

1. The Jordan canonical form of a diagonal matrix is the matrix itself.

2. Let T be a linear operator on a finite-dimensional vector space V that has a Jordan canonical form J. If  is any basis for V, then the Jordan canonical form of  is J.

3. Linear operators having the same characteristic polynomial are similar.

4. Matrices having the same Jordan canonical form are similar.

5. Every matrix is similar to its Jordan canonical form.

6. Every linear operator with the characteristic polynomial  has the same Jordan canonical form.

7. Every linear operator on a finite-dimensional vector space has a unique Jordan canonical basis.

8. The dot diagrams of a linear operator on a finite-dimensional vector space are unique.

2. Let T be a linear operator on a finite-dimensional vector space V such that the characteristic polynomial of T splits. Suppose that , and  are the distinct eigenvalues of T and that the dot diagrams for the restriction of T to  are as follows:

Find the Jordan canonical form J of T.

3. Let T be a linear operator on a finite-dimensional vector space V with Jordan canonical form

1. Find the characteristic polynomial of T.

2. Find the dot diagram corresponding to each eigenvalue of T.

3. For which eigenvalues , if any, does ?

4. For each eigenvalue , find the smallest positive integer  for which .

5. Compute the following numbers for each i, where  denotes the restriction of  to .

1. 

2. 

3. 

4. 

4. For each of the matrices A that follow, find a Jordan canonical form J and an invertible matrix Q such that . Notice that the matrices in (a), (b), and (c) are those used in Example 5.

1. 

2. 

3. 

4. 

5. For each linear operator T, find a Jordan canonical form J of T and a Jordan canonical basis  for T.

1. (a) V is the real vector space of functions spanned by the set of real-valued functions , and T is the linear operator on V defined by .

2. (b) T is the linear operator on  defined by .

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

4. (d) T is the linear operator on  defined by


5. (e) T is the linear operator on  defined by


6. (f) V is the vector space of polynomial functions in two real variables x and y of degree at most 2, as defined in Example 4, and T is the linear operator on V defined by


6. Let A be an  matrix whose characteristic polynomial splits. Prove that A and  have the same Jordan canonical form, and conclude that A and  are similar. Hint: For any eigenvalue  of A and  and any positive integer r, show that .

7. Let A be an  matrix whose characteristic polynomial splits,  be a cycle of generalized eigenvectors corresponding to an eigenvalue , and W be the subspace spanned by . Define  to be the ordered set obtained from  by reversing the order of the vectors in .

1. (a) Prove that .

2. (b) Let J be the Jordan canonical form of A. Use (a) to prove that J and  are similar.

3. (c) Use (b) to prove that A and  are similar.

8. Let T be a linear operator on a finite-dimensional vector space, and suppose that the characteristic polynomial of T splits. Let  be a Jordan canonical basis for T.

1. (a) Prove that for any nonzero scalar c,  is a Jordan canonical basis for T.

2. (b) Suppose that  is one of the cycles of generalized eigenvectors that forms , and suppose that  corresponds to the eigenvalue  and has length greater than 1. Let x be the end vector of , and let y be a nonzero vector in . Let  be the ordered set obtained from  by replacing x by . Prove that  is a cycle of generalized eigenvectors corresponding to , and that if  replaces  in the union that defines , then the new union is also a Jordan canonical basis for T.

3. (c) Apply (b) to obtain a Jordan canonical basis for , where A is the matrix given in Example 2, that is different from the basis given in the example.

9. Suppose that a dot diagram has k columns and m rows with  dots in column j and  dots in row i. Prove the following results.

1. (a)  and .

2. (b)  for  and  for . Hint: Use mathematical induction on m.

3. (c) 

4. (d) Deduce that the number of dots in each column of a dot diagram is completely determined by the number of dots in the rows.

10. Let T be a linear operator whose characteristic polynomial splits, and let  be an eigenvalue of T.

1. (a) Prove that  is the sum of the lengths of all the blocks corresponding to  in the Jordan canonical form of T.

2. (b) Deduce that  if and only if all the Jordan blocks corresponding to  are  matrices.

The following definitions are used in Exercises 11–19.

# Definitions.

A linear operator T on a vector space V is called nilpotent if  for some positive integer p. An  matrix A is called nilpotent if  for some positive integer p.

1. Let T be a linear operator on a finite-dimensional vector space V, and let  be an ordered basis for V. Prove that T is nilpotent if and only if  is nilpotent.

2. Prove that any square upper triangular matrix with each diagonal entry equal to zero is nilpotent.

3. Let T be a nilpotent operator on an n-dimensional vector space V, and suppose that p is the smallest positive integer for which . Prove the following results.

1. (a)  for every positive integer i.

2. (b) There is a sequence of ordered bases  such that  is a basis for  and  contains  for .

3. (c) Let  be the ordered basis for  V in (b). Then  is an upper triangular matrix with each diagonal entry equal to zero.

4. (d) The characteristic polynomial of T is . Hence the characteristic polynomial of T splits, and 0 is the only eigenvalue of T.

4. Prove the converse of Exercise 13(d): If T is a linear operator on an n-dimensional vector space V and  is the characteristic polynomial of T, then T is nilpotent.

5. Give an example of a linear operator T on a finite-dimensional vector space over the field of real numbers such that T is not nilpotent, but zero is the only eigenvalue of T. Characterize all such operators. Visit goo.gl/nDjsWm for a solution.

6. Let T be a nilpotent linear operator on a finite-dimensional vector space V. Recall from Exercise 13 that  is the only eigenvalue of T, and hence . Let  be a Jordan canonical basis for T. Prove that for any positive integer i, if we delete from  the vectors corresponding to the last i dots in each column of a dot diagram of , the resulting set is a basis for . (If a column of the dot diagram contains fewer than i dots, all the vectors associated with that column are removed from .)

7. Let T be a linear operator on a finite-dimensional vector space V such that the characteristic polynomial of T splits, and let  be the distinct eigenvalues of T. For each i, let  denote the unique vector in  such that . (This unique representation is guaranteed by Theorem 7.3 (p. 479).) Define a mapping  by


1. (a) Prove that S is a diagonalizable linear operator on V.

2. (b) Let . Prove that U is nilpotent and commutes with S, that is, .

8. Let T be a linear operator on a finite-dimensional vector space V over C, and let J be the Jordan canonical form of T. Let D be the diagonal matrix whose diagonal entries are the diagonal entries of J, and let . Prove the following results.

1. (a) M is nilpotent.

2. (b) 

3. (c) If p is the smallest positive integer for which , then, for any positive integer ,



and, for any positive integer ,


9. Let  and



be the  Jordan block corresponding to , and let . Prove the following results:

1. (a) , and for ,


2. (b) For any integer ,


3. (c)  exists if and only if one of the following holds:

1. .

2.  and .

(Note that  exists under these conditions. See the discussion preceding Theorem 5.12 on page 284.) Furthermore,  is the zero matrix if condition (i) holds and is the  matrix (1) if condition (ii) holds.

4. (d) Prove Theorem 5.12 on page 284.

The following definition is used in Exercises 20 and 21.

# Definition.

For any , define the norm of A by


1. Let . Prove the following results.

1. (a) .

2. (b)  if and only if .

3. (c)  for any scalar c.

4. (d) .

5. (e) 

2. Let  be a transition matrix. (See Section 5.3.) Since C is an algebraically closed field, A has a Jordan canonical form J to which A is similar. Let P be an invertible matrix such that . Prove the following results.

1. (a)  for every positive integer k.

2. (b) There exists a positive number c such that  for every positive integer k.

3. (c) Each Jordan block of J corresponding to the eigenvalue  is a  matrix.

4. (d)  exists if and only if 1 is the only eigenvalue of A with absolute value 1.

5. (e) Theorem 5.19(a), using (c) and Theorem 5.18.

The next exercise requires knowledge of absolutely convergent series as well as the definition of  for a matrix A. (See page 310.)

1. Use Exercise 20(d) to prove that  exists for every .

2. Let  be a system of n linear differential equations, where x is an n-tuple of differentiable functions  of the real variable t, and A is an  coefficient matrix as in Exercise 16 of Section 5.2. In contrast to that exercise, however, do not assume that A is diagonalizable, but assume that the characteristic polynomial of A splits. Let  be the distinct eigenvalues of A.

1. (a) Prove that if u is the end vector of a cycle of generalized eigenvectors of  of length p and u corresponds to the eigenvalue , then for any polynomial f(t) of degree less than p, the function



is a solution to the system .

2. (b) Prove that the general solution to  is a sum of the functions of the form given in (a), where the vectors u are the end vectors of the distinct cycles that constitute a fixed Jordan canonical basis for .

3. Use Exercise 23 to find the general solution to each of the following systems of linear equations, where x, y, and z are real-valued differentiable functions of the real variable t.

1. 

2.