5.5 Application to a Single Linear Transformation

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Let V be a finite dimensional vector space over the field F, and let T be

a linear transformation on V . Using the above methods, we shall be able to

compute the so-called rational canonical form of the transformation T.

The first important step is to regard V an an F[x]-module, where x

is an indeterminate. Indeed, simply take the scalar multiplication to be

f(x)·v = f(T)(v), where f(x) 2 F[x], v 2 V . This is easily checked to satisfy

the requirements of a scalar multiplication. Note that since F[x] is a principal

ideal domain, the results of the preceeding section apply. The following is

quite simple, and provides the existence of the minimal polynomial of the

linear transformation T:

Lemma 5.5.1 The F[x]-module V defined above is a finitely generated torsion

module.

Indeed, the exponent of the F[x]-module V is nothing other than the

minimal polynomial of T.

Recall that the idea behind canonical forms for a linear transformation

T is to find a basis for V relative to which T has a particularly simple form.

Since the structure theory for finitely generated torsion modules over a p.i.d.

rests on a decomposition into cyclic modules, it is appropriate to investigate

first what happens when the F[x]-module V is itself cyclic. The answer is

provided below.

Lemma 5.5.2 Let V be an n-dimensional F-vector space with linear transformation

T 2 EndF(V ), and assume that the F[x]-module V is cyclic. Then

there exists a basis of V with respect to which T is represented by the matrix

A =

2

6666664

0 0 . . −a0

1 0 . . .

0 1 . . .

. . . . .

. . . 0 −an−2

. . . 1 −an−1

3

7777775

,

where f(x) =

Xn

i=0

aixi is the exponent of the F[x]-module V .

124 CHAPTER 5. MODULE THEORY

The matrix above is called the companion matrix of the polynomial f(x).

Let V have basis {v1, v2, . . . , vn}, and let T 2 EndF(V ). Assume that

T(vi) =

P

_jivj , i = 1, 2 . . . , n. Let F be the free F[x]-module with basis

{e1, e2, . . . , en}; there is an F[x]-module homomorphism F ! V with ei 7!

vi, i = 1, 2, . . . , n.

Lemma 5.5.3 If K = ker(F ! V ), then the elements

fi = xei −

Xn

j=1

_jiej , i = 1, 2, . . . , n,

generate K.

Note that by Theorem 5.2.9, together with Exercise 2 Section 5.3, we see

that the above elements f1, . . . , fn actually comprise a basis of K. However,

this fact is important for our purposes.

From the above theorem, we see that the matrix (xI − A)t, where

A = [_ij ], represents the linear transformation T 2 End(V ), is the relations

matrix for the presentation of the F[x]-module as a quotient of a free

module. Furthermore, by Exercise 3 of Section 5.4, we see that in order to

find the invariant factors of V as an F[x]-module (i.e., the invariant factors

of the linear transformation T), it suffices to compute the Smith canonical

form of the matrix (xI−A) 2 Mn(F[x]). Therefore, if f1(x), f2(x), . . . , fr(x)

are the invariant factors, then there is a basis of V with respect to which T

is represented by the block diagonal matrix

A =

2

66664

C1 0 . 0

0 C2 . 0

0 0 . .

. . .

0 0 . Cr

3

77775

,

where Ci is the companion matrix of fi(x), i = 1, 2, . . . , r. The above matrix

form is called the rational canonical form of the linear transformation T 2

End(V ).

Furthermore, note that each invariant factor fi(x) divides det (xI − A),

i.e., each invariant factor divides the characteristic polynomial of the linear

transformation T. In particular, one has

Theorem 5.5.4 (Cayley-Hamilton Theorem) Let T be a linear transformation

on the finite-dimensional vector space V . If mT (x) and cT (x)

5.5. APPLICATION TO A SINGLE LINEAR TRANSFORMATION 125

denote the minimal polynomial and characteristic polynomial, respectively,

of T, then mT (x) cT (x).

As a simple example, we consider the transformation represented by the

matrix

A =

2

4

5 −8 4

6 −11 6

6 −12 7

3

5.

Note that det(xI −A) = (x−1)2(x+1); thus the invariant factors of A are

divisors of (x−1)2(x+1) (see Theorem 5.5.4, above). Let’s compute them.

After some work, one arrives at the Smith canonical form

A =

2

4

1 0

0 x − 1

0 0 (x − 1)(x + 1)

3

5,

from which it follows that the rational canonical form for A is

A =

2

4

1 0 0

0 0 1

0 1 0

3

5.

As another type of example, suppose that we have a matrix A, taken over

the rational field, whose determinant is cA(x) = (x − 1)2(x2 − x + 1)(x2 +

x + 1)3. Thus A is a 10 × 10 matrix. We may list the possible invariant

factors below

1) (x − 1)2(x2 − x + 1)(x2 + x + 1)3

2) (x − 1), (x − 1)(x2 − x + 1)(x2 + x + 1)3

3) (x2 + x + 1), (x − 1)2(x2 − x + 1)(x2 + x + 1)2

4) (x − 1)(x2 + x + 1), (x − 1)(x2 − x + 1)(x2 + x + 1)2

5) (x2 + x + 1), (x2 + x + 1), (x − 1)2(x2 − x + 1)(x2 + x + 1)

6) (x2 + x + 1), (x − 1)(x2 + x + 1), (x − 1)(x2 − x + 1)(x2 + x + 1)

(The reader is encouraged to find all possible sets of elementary divisors.)

Finally, we give a brief development of the so-called Jordan canonical

form for a linear transformation T : V ! V , where V is finite dimensional

126 CHAPTER 5. MODULE THEORY

over the field F. Here, however, we need to assume that the minimal polynomial

splits completely into linear factors in F[x]. Thus assume that mT (x) =

(x − _1)e1(x − _2)e2 · · · (x − _r)er , with _1, _2, . . . , _r 2 F. By the Primary

Decomposition Theorem, we may as well assume that mT (x) = (x − _)e.

Note that if we set T0 = T − _, them m0

T (x) = xe; thus there exists a basis

of V with respect to which T0 is represented by a block diagonal matrix,

whose diagonal blocks are of the form

2

66664

0 0 . . 0

1 0 . . 0

0 1 . . .

. . . . .

0 0 . 1 0

3

77775

.

From the above, we conclude that the original linear transformation T is

represented by a block diagonal matrix, whose blocks are “Jordan blocks”

of the form

Jk(_) =

2

66664

_ 0 . . 0

1 _ . . 0

0 1 . . .

. . . . .

0 0 . 1 _

3

77775

,

where the index k above simply means that Jk(_) is a k × k matrix. The

above representation of the linear transformation T as a block diagonal

matrix consisting of Jordan blocks is called the Jordan canonical form of T.

Exercises

1. Find the rational canonical form for the matrix

A =

2

664 3

−

1

1

−

1

0 3 −1 1

2 −1 3 −4

3 −3 3 −4

3

775

.

2. Do the same for

A =

2

664

6 2 3 0

2 3 −4 1

−3 3 1 2

−1 2 −3 5

3

775

.

5.5. APPLICATION TO A SINGLE LINEAR TRANSFORMATION 127

3. Let A be a rational coefficient matrix with minimal polynomial mA(x) =

(x+2)2(x2+1)2(x4−x2+1). If A is a 16×16 matrix, find the possible

lists of invariant factors.

4. Let A,B be n × n matrices over the field F, and let K _ F. If A,B

are similar over K, prove that they are similar over F.

5. Let V be a finite dimensional vector space over the field F, and let

T : V ! V be a linear transformation. Assume that mT (x) = p(x) 2

F[x], where p(x) is irreducible. If we set K = F[x]/(p(x)), show that

V can be regarded in a natural way as a K-vector space in such a way

that the K-subspaces of V are in bijective correspondence with the

T-invariant F-subspaces of V . (Hint: define K-scalar multiplication

by setting (f(x) + I) · v = f(t)(v), f(x) 2 F[x], and where I is the

principal ideal I = (p(x)).)

6. Let V be a finite dimensional vector space over the field F, and let

T : V ! V be a linear transformation. Say that T is semisimple if and

only every T-invariant subspace W _ V has a T-invariant subspace

U _ V with V = W _ U. Prove that if the minimal polynomial of

T factors into the product of distinct irreducible factors in F[x], then

T is semisimple. (Hint: Let V = V1 _ V2 _ · · · _ Vr be the primary

decomposition of V , and let W _ V be a T-invariant subspace of V .

Argue that W = (W \ V1) _ (W \ V2) _ · · · _ (W \ Vr), and apply

Exercise 5 to each component.)

7. Let Fq be the finite field of order q, and let G = GL2(q).

(a) Show that for every _ 2 G, the minimal polynomial m_(x) splits

over Fq2 .

(b) Write down the conjugacy classes of G with representatives written

in terms of their Jordan canonical forms over Fq2 .

8. Let F = R (real field). If A 2 Mn(R), define eA =

P1

k=0 Ak.

(a) Prove that

P1

k=0 Ak is an absolutely convergent series; thus eA

is well-defined for any matrix A 2 Mn(R)

(b) If A,B 2 Mn(R) with AB = BA, prove that eA+B = eAeB.

(c) If J = J3(_) (3 × 3 Jordan block), compute eJ .

128 CHAPTER 5. MODULE THEORY

(d) In general, describe a procedure for computing eA, for any matrix

A 2 Mn(R), in terms of matrices P, J, where P−1AP = J, and

where J is a matrix in Jordan canonical form.

5.6. CHAIN CONDITIONS AND SERIES OF MODULES 129

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