0.1 Rational Canonical Forms

Draft. Do not cite or quote.

We have already seen that it is useful and simpler to study linear systems using matrices. But matrices are themselves cumbersome, as they are stuffed with many entries, and it turns out that it’s best to focus on the fact that we can view an m × n matrix A as a function n m n T : R → R ; namely, T (x) = Ax for every column vector x ∈ R . The function T , called a linear transformation, has two important properties: T (x + y) = T (x) + T (y), for all x, y; for every number α and vector x, we have T (αx) = αT (x). As we shall see in the course, the matrix A is a way of studying T by choosing a coordinate n system in R ; choosing another coordinate system gives another matrix describing T . Just as in calculus when studying conic sections, choosing a good coordinate system can greatly simplify things: we can put a conic section defined by Ax2 + Bxy + Cy2 + Dx + Ey + F into normal form, and we can easily read off properties of the conic (is it a parabola, hyperbola, or ellipse; what are its foci, etc) from the normal form. The important object of study for us is a linear transformation T , and any choice of coordinate system produces a matrix describing it. Is there a “best” matrix describing T analogous to a normal form for a conic? The answer is yes, and normal forms here are called canonical forms. Now coordinates arise from bases of a vector space, and the following discussion is often called change of basis.

0.1 Rational Canonical Forms

We begin by stating some facts which will eventually be treated in the course proper. If you have any questions, please feel free to ask me. Unless we say otherwise, our vector spaces involve real numbers as scalars (although, in several instances, we may want complex numbers), linear transformations T are functions on a vector space V (that is, T maps V to itself), and matrices are square. If T : V → V is a linear transformation on a vector space V and x = x1, . . . , xn is a basis of V , then T determines the matrix A = x[T ]x whose ith column consists of the coordinate list of T (xi) with respect to x. If Y is another basis of V , then the matrix B = Y [T ]Y may be diﬀerent from A, but Theorem 4.3.1 in the book says that A and B are similar; that is, there exists a nonsingular matrix P with B = P AP −1.

Theorem 4.3.1. Let T : V → V be a linear transformation on a vector space V . If x and Y are bases of V , then there is a nonsingular

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matrix P (called a transition matrix), namely, P = Y [1V ]x, so that −1 Y [T ]Y = P x[T ]x P .

Conversely, if B = P AP −1, where B,A, and P are n × n matrices n and P is nonsingular, then there is a linear transformation T : R → n n R and bases x and Y of k such that B = Y [T ]Y and A = x[T ]x.

We now consider how to determine whether two given matrices are similar.

Deﬁnition If A is an r × r matrix and B is an s × s matrix, then their direct sum A ⊕ B is the (r + s) × (r + s) matrix " # A 0 A ⊕ B = . 0 B

Deﬁnition

If g(x) = x + c0, then its companion matrix C(g) is the 1 × 1 s s−1 matrix [−c0]; if s ≥ 2 and g(x) = x + cs−1x + ··· + c1x + c0, then its companion matrix C(g) is the s × s matrix   0 0 0 ··· 0 −c0 1 0 0 ··· 0 −c   1    0 1 0 ··· 0 −c2  C(g) =   . 0 0 1 ··· 0 −c3    ......  ......  0 0 0 ··· 1 −cs−1

Obviously, we can recapture the polynomial g(x) from the last column of the companion matrix C(g). We call a polynomial f(x) monic if the highest power of x in f has coeﬃcient 1.

Theorem 0.1 Every n×n matrix A is similar to a direct sum of companion matrices

C(g1) ⊕ · · · ⊕ C(gt)

in which the gi(x) are monic polynomials and

g1(x) | g2(x) | · · · | gt(x).

Deﬁnition

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A rational canonical form is a matrix R that is a direct sum of companion matrices,

R = C(g1) ⊕ · · · ⊕ C(gt),

where the gi(x) are monic polynomials with g1(x) | g2(x) | · · · | gt(x).

If a matrix A is similar to a rational canonical form C(g1) ⊕ · · · ⊕ C(gt), where g1(x) | g2(x) | · · · | gt(x), then we say that the invariant factors of A are g1(x), g2(x), . . . , gt(x).

Theorem 0.1 says that every n × n matrix is similar to a rational canonical form, and so it has invariant factors. Can a matrix A have more than one list of invariant factors?

Theorem 0.2 1. Two n × n matrices A and B are similar if and only if they have the same invariant factors. 2. An n × n matrix A is similar to exactly one rational canonical form.

Corollary 0.3 Let A and B be n × n matrices with real entries. If A and B are similar over C, then they are similar over R (i.e., if there is a nonsingular matrix P having complex entries with B = P AP −1, then there is a nonsingular matrix Q having real entries with B = QAQ−1).

Deﬁnition If T : V → V is a linear transformation, then an invariant subspace is, a subspace W of V with T (W ) ⊆ W .

Does a linear transformation T on a ﬁnite-dimensional vector space V leave any one-dimensional subspaces of V invariant; that is, is there a nonzero vector v ∈ V with T (v) = αv for some α? If 2 2 o T : R → R is rotation by 90 , then its matrix with respect to the 0 −1 standard basis is 1 0 . Now " # " #" # " # x 0 −1 x −y T : 7→ = . y 1 0 y x

x If v = [ y ] is a nonzero vector and T (v) = αv for some α ∈ R, then αx = −y and αy = x; it follows that (α2 +1)x = x and (α2 +1)y = y. 2 Since v 6= 0, α + 1 = 0 and α∈ / R. Thus, T has no one-dimensional

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0 −1 invariant subspaces. Note that 1 0 is the companion matrix of x2 + 1.

Deﬁnition Let V be a vector space and let T : V → V be a linear transformation. If T v = αv, where α ∈ C and v ∈ V is nonzero, then α is called an eigenvalue of T and v is called an eigenvector of T for α. Let A be an n × n matrix. If Av = αv, where α ∈ k and v ∈ kn is a nonzero column, then α is called an eigenvalue of A and v is called an eigenvector of A for α.

Rotation by 90o has no (real) eigenvalues. At the other extreme, can a linear transformation have inﬁnitely many eigenvalues?

n n Theorem 4.2.1. If T : R → R is a linear transformation, then there exists a unique n × n matrix A such that T (v) = Av for all n v ∈ R .

To say that Av = αv for v nonzero is to say that v is a nontrivial solution of the homogeneous system (A−αI)v = 0; that is, A−αI is a singular matrix. But a matrix is singular if and only if its determinant is 0.

Deﬁnition The characteristic polynomial of an n × n matrix A is

pA(x) = det(xI − A) ∈ R[x].

Thus, the eigenvalues of an n × n matrix A are the roots of pA(x), a polynomial of degree n, and so A has at most n real eigenvalues. Note that some eigenvalues of A may be complex numbers.

Deﬁnition

The trace of an n × n matrix A = [aij] is n X tr(A) = aii. i=1

Proposition 0.4

If A = [aij] is an n × n matrix having eigenvalues (with multiplicity) α1, . . . , αn, then X Y tr(A) = − αi and det(A) = αi. i i

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Proof. For any polynomial f(x) with real coeﬃcients, if f(x) = n n−1 x + cn−1x + ··· + c1x + c0 = (x − α1) ··· (x − αn), then cn−1 = P n Q Qn − i αi and c0 = (−1) i αi. In particular, pA(x) = i=1(x − αi), P so that cn−1 = − i αi = −tr(A). Now the constant term of any polynomial f(x) is just f(0); setting x = 0 in pA(x) = det(xI − A) n Q gives pA(0) = det(−A) = (−1) det(A). Hence, det(A) = i αi. Here are some elementary facts about eigenvalues.

Corollary 0.5 Let A be an n × n matrix.

1. A is singular if and only if 0 is an eigenvalue of A. 2. If α is an eigenvalue of A, then αn is an eigenvalue of An. 3. If A is nonsingular and α is an eigenvalue of A, then α 6= 0 and α−1 is an eigenvalue of A−1.

Proof.

1. If A is singular, then the homogeneous system Ax = 0 has a nontrivial solution; that is, there is a nonzero v with Av = 0. But this just says that Av = 0x (here, 0 is a scalar), and so 0 is an eigenvalue. Conversely, if 0 is an eigenvalue, then 0 = det(0I − A) = ± det(A), so that det(A) = 0 and A is singular. 2. There is a nonzero vector v with Av = αv. It follows by induction on n ≥ 1 that Anv = αnv. 3. If v is an eigenvector of A for α, then

v = A−1Av = A−1αv = αA−1v.

Therefore, α 6= 0 (because eigenvectors are nonzero) and α−1v = A−1v.

Let us return to rational canonical forms.

Lemma 0.6 If C(g) is the companion matrix of g(x) ∈ k[x], then

detxI − C(g) = g(x).

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Proof. If g(x) = x + c0, then C(g) is the 1 × 1 matrix [−c0], and det(xI − C(g)) = x + c0 = g(x). If deg(g) = s ≥ 2, then   x 0 0 ··· 0 c0 −1 x 0 ··· 0 c   1    xI − C(g) =  0 −1 x ··· 0 c2  ,    ......   ......  0 0 0 · · · −1 x + cs−1

and Laplace expansion across the ﬁrst row gives

1+s det(xI − C(g)) = x det(L) + (−1) c0 det(M),

where L is the matrix obtained by erasing the top row and ﬁrst column, and M is the matrix obtained by erasing the top row and last column. Now M is a triangular (s − 1) × (s − 1) matrix having −1’s on the diagonal, while L = xI −C (g(x)−c0)/x . By induction, s−1 det(L) = (g(x) − c0)/x, while det(M) = (−1) . Therefore,

(1+s)+(s−1) det(xI − C(g)) = x[(g(x) − c0)/x] + (−1) c0 = g(x).

If R = C(g1) ⊕ · · · ⊕ C(gt) is a rational canonical form, then xI − R = xI − C(g1) ⊕ · · · ⊕ xI − C(gt) .

Given square matrices B1,...,Bt, we have det(B1 ⊕ · · · ⊕ Bt) = Qt i=1 det(Bi). With Lemma 0.6, this gives

t t Y Y pR(x) = pC(gi)(x) = gi(x). i=1 i=1 Thus, the characteristic polynomial is the product of the invariant factors.

Example We now show that similar matrices have the same characteristic polynomial. If B = P AP −1, then since xI commutes with every matrix, we have P xIP −1 = (xI)PP −1 = xI. Therefore,

−1 −1 pB(x) = det(xI − B) = det(P xIP − P AP ) = det(P [xI − A]P −1) = det(P ) det(xI − A) det(P −1)

= det(xI − A) = pA(x).

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It follows that if A is similar to C(g1) ⊕ · · · ⊕ C(gt), then t Y pA(x) = gi(x). i=1 Therefore, similar matrices have the same eigenvalues with multiplicities. Hence, Proposition 0.4 says that similar matrices have the same trace and the same determinant.

Theorem 0.7 (Cayley–Hamilton)

If A is an n × n matrix with characteristic polynomial pA(x) = n n−1 x + bn−1x + ··· + b1x + b0, then pA(A) = 0; that is, n n−1 A + bn−1A + ··· + b1A + b0I = 0.

Proof. Birkhoﬀ–Mac Lane, A Survey of Modern Algebra, p. 341.

Deﬁnition

The minimal polynomial mA(x) of an n×n matrix A is the monic polynomial f(x) of least degree with the property that f(A) = 0.

Proposition 0.8

The minimal polynomial mA(x) is a divisor of the characteristic polynomial pA(x), and every eigenvalue of A is a root of mA(x).

Proof. By the Cayley–Hamilton Theorem, pA(A) = 0.

Now gt(x) is the minimal polynomial of A, where gt(x) is the invariant factor of A of highest degree. It follows from the fact that

pA(x) = g1(x) ··· gt(x),

where g1(x) | g2(x) | · · · | gt(x), that mA(x) = gt(x) is a polynomial having every eigenvalue of A as a root [of course, the multiplicity of a root of mA(x) may be less than its multiplicity as a root of the characteristic polynomial pA(x)].

Corollary 0.9

If all the eigenvalues of an n×n matrix A are distinct, then mA(x) = pA(A); that is, the minimal polynomial coincides with the characteristic polynomial.

Proof. This is true because every root of pA(x) is a root of mA(x).

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Exercises

1. a. How many 10 × 10 matrices A over R are there, up to similarity, with A2 = I? b. How many 10 × 10 matrices A over Fp are there, up to similarity, with A2 = I? Hint. The answer depends on the parity of p.

2. Find the rational canonical forms of     " # 2 0 0 2 0 0 1 2 A = ,B = 1 2 0 , and C = 1 2 0 . 3 4     0 0 3 0 1 2

3. If A is similar to A0 and B is similar to B0, prove that A ⊕ B is similar to A0 ⊕ B0.

0.2 Jordan Canonical Forms

Even if we know the rational canonical form of a matrix A, we may not know the rational canonical form of a power of A. For example, given a matrix A, is some power of it with Am = I? We now give a second canonical form whose powers are easily calculated.

Deﬁnition Let k be a ﬁeld and let α be a real or complex number. A 1 × 1 Jordan block is a matrix J(α, 1) = [α] and, if s ≥ 2, an s × s Jordan block is a matrix J(α, s) of the form

α 0 0 ··· 0 0 1 α 0 ··· 0 0     0 1 α ··· 0 0 J(α, s) =  ......  .  ......   ......    0 0 0 ··· α 0 0 0 0 ··· 1 α

Here is a more compact description of a Jordan block when s ≥ 2. Let L denote the s × s matrix having all entries 0 except for 1’s on the subdiagonal just below the main diagonal. With this notation, a Jordan block J(α, s) can be written as

J(α, s) = αI + L.

s Let us regard L as a linear transformation on k . If e1, . . . , es is the standard basis, then Lei = ei+1 if i < s while Les = 0. It follows

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easily that the matrix L2 is all 0’s except for 1’s on the second subdiagonal below the main diagonal; L3 is all 0’s except for 1’s on the third subdiagonal; Ls−1 has 1 in the s, 1 position, with 0’s everywhere else, and Ls = 0. Thus, L is nilpotent.

Lemma 0.10 If J = J(α, s) = αI + L is an s × s Jordan block, then for all m ≥ 1,

s−1 X m J m = αmI + αm−iLi. i i=1

Proof. Since L and αI commute (the scalar matrix αI commutes with every matrix), the ring generated by αI and L is commutative, and the Binomial Theorem applies. Finally, note that all terms involving Li for i ≥ s are 0 because Ls = 0.

Example Diﬀerent powers of L are “disjoint”; that is, if m 6= n and the i, j entry of Ln is nonzero, then the i, j entry of Lm is zero. For example, " #m " # α 0 αm 0 = 1 α mαm−1 αm and  m   α 0 0 αm 0 0    m−1 m  1 α 0 =  mα α 0  . m m−2 m−1 m 0 1 α 2 α mα α

Lemma 0.11 If g(x) = (x − α)s, then the companion matrix C(g) is similar to the s × s Jordan block J(α, s).

It follows that Jordan blocks also correspond to polynomials (just as companion matrices do); in particular, the characteristic polynomial of J(α, s) is the same as that of C((x − α)s):

s pJ(α,s)(x) = (x − α) .

Theorem 0.12 Let A be an n × n matrix with real entries. If all the eigenvalues of A are real, then A is similar to a direct sum of Jordan blocks.

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Deﬁnition A Jordan canonical form is a direct sum of Jordan blocks. If a matrix A is similar to the Jordan canonical form

J(α1, s1) ⊕ · · · ⊕ J(αr, sr),

then we say that A has elementary divisors

s1 sr (x − α1) ,..., (x − αr) .

Theorem 0.12 says that every square matrix A having entries in a ﬁeld containing all the eigenvalues of A is similar to a Jordan canonical form. Can a matrix be similar to several Jordan canonical forms? The answer is yes, but not really.

Example

Let Ir be the r × r identity matrix, and let Is be the s × s identity matrix. Then interchanging blocks in a direct sum yields a similar matrix: " # " #" #" # B 0 0 I A 0 0 I = r s . 0 A Is 0 0 B Ir 0

Since every permutation is a product of transpositions, it follows that permuting the blocks of a matrix of the form A1 ⊕ A2 ⊕ · · · ⊕ At yields a matrix similar to the original one.

Theorem 0.13 1. If A and B are n×n matrices over C, then A and B are similar if and only if they have the same elementary divisors. 2. If a matrix A is similar to two Jordan canonical forms, say, H and H0, then H and H0 have the same Jordan blocks (i.e., H0 arises from H by permuting its Jordan blocks).

The hypothesis that all the eigenvalues of A and B lie in C is not a serious problem. Recall that Corollary 0.3(ii) says that if A and B are similar over C, then they are similar over R. Here are some applications of canonical forms.

Proposition 0.14 If A is an n × n matrix, then A is similar to its transpose A>.

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Proof. First, Corollary 0.3(ii) allows us to assume that k contains all the eigenvalues of A. Now if B = P AP −1, then B> = (P >)−1A>P >; that is, if B is similar to A, then B> is similar to A>. Thus, it suﬃces to prove that H is similar to H> for a Jordan canonical form H; by Exercise 3, it is enough to show that a Jordan block J = J(α, s) is similar to J >. We illustrate the idea for J(α, 3). Let Q be the matrix having 1’s on the “wrong” diagonal and 0’s everywhere else; notice that Q = Q−1:         0 0 1 α 0 0 0 0 1 α 1 0         0 1 0 1 α 0 0 1 0 = 0 α 1 . 1 0 0 0 1 α 1 0 0 0 0 α

Let v1, . . . , vs be a basis of a vector space W , define Q: W → W by Q: vi 7→ vs−i+1, and define J : W → W by J : vi 7→ αvi + vi+1 for −1 i < s and J : vs 7→ αvs. The reader can now prove that Q = Q and QJ(α, s)Q−1 = J(α, s)>. Exponentiating a matrix is used to find solutions to systems of linear differential equations. An n × n complex matrix A consists of 2 n2 n entries, and so A may be regarded as a point in C . This allows us to define convergence of a sequence of n × n complex matrices: A1,A2,...,Ak,... converges to a matrix M if, for each i, j, the sequence of i, j entries converges. As in Calculus, convergence of a series means convergence of the sequence of its partial sums.

Deﬁnition If A is an n × n complex matrix, then ∞ X 1 eA = Ak = I + A + 1 A2 + 1 A3 + ··· . k! 2 6 k=0

This series converges for every matrix A (see Exercise 7), and the A function A 7→ e is continuous; that is, if limk→∞ Ak = M, then

lim eAk = eM k→∞

We are now going to show that the Jordan canonical form of A can be used to compute eA.

Proposition 0.15

Let A = [aij] be an n × n complex matrix.

1. If P is nonsingular, then P eAP −1 = eP AP −1 .

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2. If AB = BA, then eAeB = eA+B. 3. For every matrix A, the matrix eA is nonsingular; indeed,

(eA)−1 = e−A.

4. If L is the n×n matrix having 1’s just below the main diagonal and 0’s elsewhere, then eL is a lower triangular matrix with 1’s on the diagonal.

5. If D is a diagonal matrix, say, D = diag(α1, α2, . . . , αn), then

eD = diag(eα1 , eα2 , . . . , eαn ).

6. If α1, . . . , αn are the eigenvalues of A (with multiplicities), then eα1 ,... , eαn are the eigenvalues of eA (with multiplicities). 7. We can compute eA. 8. If tr(A) = 0, then det(eA) = 1.

Proof.

1. We use the continuity of matrix exponentiation: n X 1 P eAP −1 = P lim Ak P −1 n→∞ k! k=0 n X 1 = lim PAkP −1 n→∞ k! k=0 n X 1 k = lim P AP −1 n→∞ k! k=0 −1 = eP AP .

2. The coeﬃcient of the kth term of the power series for eA+B is 1 (A + B)k, k! while the kth term of eAeB is k k X 1 1 X 1 1 X k Ai Bj = AiBk−i = AiBk−i. i! j! i!(k − i)! k! i i+j=k i=0 i=0 Since A and B commute, the Binomial Theorem shows that both kth coeﬃcients are equal. (See Exercise 9 for an example where this is false ifA and B do not commute.) 3. This follows immediately from part (ii), for −A and A commute and e0 = I.

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4. The equation 1 1 eL = I + L + L2 + ··· + Ls−1 2 (s − 1)! holds because Ls = 0, and the result follows by Lemma 0.10. For example, when s = 5,

 1 0 0 0 0  1 1 0 0 0   L  1  e =  1 1 0 0 .  2   1 1 1 1 0  6 2  1 1 1 24 6 2 1 1

5. This is clear from the deﬁnition:

D 1 2 1 3 e = I + D + 2 D + 6 D + ··· , k k k k for D = diag(α1, α2, . . . , αn). 6. Since C is algebraically closed, A is similar to its Jordan canonical form J: there is a nonsingular matrix P with P AP −1 = J. Now A and J have the same characteristic polynomial and, hence, the same eigenvalues with multiplicities. But J is a lower triangular matrix with the eigenvalues α1, . . . , αn of A on the diagonal, and so the deﬁnition of matrix exponentiation gives eJ lower triangular with eα1 , . . . , eαn on the diagonal. Since eA = eP −1JP = P −1eJ P , it follows that the eigenvalues of eA are as claimed. 7. By the Jordan Decomposition (Exercise 2), there is a nonsingular matrix P with P AP −1 = ∆ + L, where ∆ is a diagonal matrix, Ln = 0, and ∆L = L∆. Hence,

−1 P eAP −1 = eP AP = e∆+L = e∆eL.

But e∆ is computed in part (v) and eL is computed in part (iv). Hence, eA = P −1e∆eLP is computable. 8. By Proposition 0.4, −tr(A) is the sum of its eigenvalues, while det(A) is the product of the eigenvalues. Since the eigenvalues of eA are eα1 , . . . , eαn , we have

A Y α P α −tr(A) det(e ) = e i = e i i = e . i Hence, tr(A) = 0 implies det(eA) = 1.

Exercises

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1. Find all n × n matrices A over a ﬁeld k for which A and A2 are similar.

2. (Jordan Decomposition) Prove that every n × n complex matrix A can be written as

A = D + N,

where D is diagonalizable (i.e., D is similar to a diagonal matrix), N is nilpotent (i.e., N m = 0 for some m ≥ 1), and DN = ND.

3. Give an example of an n × n complex matrix that is not diagonalizable. [It is known that every hermitian matrix A is diagonalizable (A is hermitian if A = A∗, where the i, j ∗ entry of A is aji).] Hint. A rotation (not the identity) about 2 the origin on R sends no line through the origin into itself. 4. a. Prove that all the eigenvalues of a nilpotent matrix are 0. b. Use the Jordan form to prove the converse: if all the eigenvalues of a matrix A are 0, then A is nilpotent. (This result also follows from the Cayley–Hamilton Theorem.)

5. How many similarity classes of 6 × 6 nilpotent matrices are there over a ﬁeld k?

6. If A and B are similar and A is nonsingular, prove that B is nonsingular and that A−1 is similar to B−1.

7. Let A = [aij] be an n × n complex matrix. s a. If M = maxij |aij|, prove that no entry of A has absolute value greater than (nM)s. b. Prove that the series deﬁning eA converges. c. Prove that A 7→ eA is a continuous function.

8. a. Prove that every nilpotent matrix N is similar to a strictly lower triangular matrix (i.e., all entries on and above the diagonal are 0). b. If N is a nilpotent matrix, prove that I+N is nonsingular.

1 0 0 1 A B B A 9. Let A = [ 0 0 ] and B = [ 0 0 ]. Prove that e e 6= e e and eAeB 6= eA+B.

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