0387986995Basic TheoryC

is said to be upper-triangular if ajk = 0 for j > k. The following lemma is a basic result in the theory of matrices.

Lemma IV-1-1. For each A E Mn(C), there exists a matrix P E GL(n,C) such that P-1 AP is uppertriangular.

Proof.

Let A be an eigenvalue of A and pt be an eigenvector of A associated with the eigenvalue A. Then, Apt = A P t and P 1 # 0. Choose n - 1 vectors p", (j = 2, ... , n) so that Q = [#1A p"nJ E GL(n, C), where the p"j are column vectors of the matrix

A matrix A E Mn(C) is said to be diagonal if a,k = 0 for j k. We denote by diag(di, d2, ... , d, the diagonal matrix with entries dl, d2, ... , do on the main diagonal (i.e., d_, = aj.,). A matrix A E Mn(C) is said to be diagonalizable (or semisimple) if there exists a matrix P E GL(n,C) such that P-1AP is diagonal.

Denote by Sn the set of all diagonalizable matrices in Mn(C). The following lemma is another basic result in the theory of matrices.

Lemma IV-1-2. A matrix A E Mn(C) is diagonalizable if and only if A has n linearly independent eigenvectors pt, p2, ... , Pn

Proof

If A has n linearly independent eigenvectors pt, p'2, ... , p,, set P = [P'tpa ... pnJ

E GL(n,C). Then, P'1AP is diagonal. Conversely, if PAP is diagonal for

P = [ 1 6 t h... fl, E GL(n, C), then p1, A, ... , pn 7are n linearly independent eigenvectors of A.

In particular, if a matrix A E Mn(C) has n distinct eigenvalues, then n eigenvectors corresponding to these n eigenvalues, respectively, are linearly independent (cf.

[Rab, p. 1861). Therefore, we obtain the following corollary of Lemma IV-1-2.

Corollary IV-1-3. If a matrix A E Mn(C) has n distinct eigenvalues, then A E

Sn.

The set Mn(C) is a noncommutative C-algebra. This means that Mn(C) is a vector space over C and a noncommutative ring. The set Sn is not a subalgebra of

Mn(C). However, the following lemma shows an important topological property of

Sn as a subset of Mn(C).

Lemma IV-1-4. The set Sn is dense in Mn(C).

Prioof.

It must be shown that, for each matrix A E Mn(C), there exists a sequence

may assume without any loss of generality that A is an upper-triangular matrix with the eigenvalues At, ... , An on the main diagonal (cf. Lemma IV-1-1). Set

Lemma IV-1-8. A matrix N E .Mn(C) is nilpotent if and only if then exists a matrix P E GL(n,C) such that P-I NP is upper-triangular and the entries on the main diagonal of N are all zero. Furthermore, if N is a real matrix, then there exists a real matrix P that satisfies the requirement given above.

To verify the last statement of this lemma, use a method similar to the proof of Lemma IV-1-1 together with the fact that if an eigenvalue of a real matrix is real, then there exists a real eigenvector associated with this real eigenvalue. Details are left to the reader as an exercise.

The main concern of this section is to explain the S-N decomposition of a matrix

A E Mn(C) (cf. Theorem IV-1-11). Before introducing the S-N decomposition, we need some preparation.

Let A. (j = 1, 2,... , k) be the distinct eigenvalues of A and let m., (j =

1,2.... , k) be their respective multiplicities. Then, the characteristic polynomial of the matrix A is given by pA(A) = (A - '\0M'(1\ - A2)m2 ... (A - Ak)m'. Decom-

quantity Q, is a nonzero polynomial in A of degree not greater than m, - 1. Hence,

k

I = EQ,(A) J1 (A - Ah)m". Setting

)=1 h¢j

P,(A) = Qj(A) 1I (A - A,)me

h=1

In the following two lemmas, we show that (IV.1.5) is a resolution of the identity in terms of projections Ph (A) onto invariant subspaces of A associated with eigenvalues Ah, respectively.

Lemma IV-1-9. The k matrices P, (A) (j = 1,2,... , k) given by (W-1.4) satisfy the following conditions:

(i) A and P, (A) (y = 1, 2, ... , k) commute.

(ii) (A-),In)'n'Pi(A) =0 (j = 1,2,... ,k),

(iii) P,(A)Ph(A) = 0 ifj h,

k

(iv) >Ph(A) = In,

h=1

(v) Pi(A)2=P,(A) (j=1.2,...,k),

( v : ) P, (A) 0 (j = 1, 2, ... , k).

Proof.

Since P,(A) is a polynomial of A, we obtain (i). Using Theorem IV-1-5, we derive (ii) and (iii) from (IV.1.4) and (i). Statement (iv) is the same as (IV.1.5).

Multiplying the both sides of (IV.1.5) by P,(A), we obtain

h=1

Then, (v) follows from (IV.1.6) and (iii). To prove (vi), let IT, be an eigenvector of A associated with the eigenvalue Al. Note that (IV.1.2) implies Ph(A)) = 0 if h 0 j.

Therefore, we derive P,(A,) = 1 from (IV.1.3). Now, since P. (A)#, = P,(A3)p' # we obtain (vi).

Lemma IV-1-10. Denote by V. the image of the mapping P,(A) : C" -. Cn.

Then,

(1)p'E Cn belongs to V. if and only if P,(A)p= p

(2)Pj(A)p"=0 for all fl E Vh if j 0 h.

(4)for each j, V, is an invariant subspace of A.

(5)the restriction of A on V, has a coordinates-wise representation:

where I. is the identity matrix and Nj is a nilpotent matrix.

(6) dime V, = m, .

Proof

Each part of this lemma follows from Lemma IV-1-9 as follows.

A vector IT E V3 if and only if p" = P, (A)q" for some q' E Cn. If p" = P, (A) q, we obtain P,(A)p= Pj(A)2q'= Pj(A)q =p""from (v) of Lemma IV-1-9.

A vector p" E Vh if and only if ff = Ph (A),y for some q" E Cn. Hence, from (iii) of Lemma IV-1-9 we obtain P, (A)IF = Pj (A)Ph(A)q"= 0 if 0 h.

for some g,E V2 (j = 1,2,... , k), then, by (1) and (2), we obtain P,(A)p" =

Pi(A)p1+...+PJ(A)pk =pj

Ap"= AP,(A)p = P,(A)Ap E V3 for every 15E Vj.

Let n, be the dimension of the space V, over C and let {ff,,t : 1 = 1, 2, ... , n. } be a basis for V,. Then, there exists an n. X n, matrix N,, such that

from Lemmas IV-1-9 and IV-I-10 and (IV.1.10). Hence, S is diagonalizable and N is nilpotent. Furthermore, NS = SN since S and N are polynomials in A. This shows the existence of S and N satisfying (a), (b), (c), and (d). Moreover, from

(IV.1.4), it follows that two matrices S and N are polynomials in A with coefficients in the field Q(ajk, Ah).

Step 2. Uniqueness of S and N. Assume that there exists another pair (S, N) of n x n matrices satisfying conditions (a), (b), (c), and (d). Then, (c) and (d) imply that SA = AS and NA = AN. Hence, SS = SS, NS = SN, SN = NS, and NN = NN since S and N are polynomials in A. This implies that S - S is

diagonalizable and N - N is nilpotent. Therefore, from S - S = N - N, it follows

that S-S=N-N=O.

Step 3. The case when S and N are real. In case when A is real, let 5 and N be the complex conjugates of S and N, respectively. Then, A = S + N = 3° + N.

Hence, the uniqueness of S and N implies that S = 3 and N = N.

This completes the proof of Theorem IV-1-11.

Definition IV-1-12. The decomposition A = S + N of Theorem IV-1-11 is called the S-N decomposition of A.

Remark IV-1-13. From (IV.1.11), it follows immediately that S and A have the same eigenvalues, counting their multiplicities. Therefore, S is invertible if and only if A is invertible.

Observation IV-1-14. Let A be an n x n matrix whose distinct eigenvalues are A = S + N be the S-N decomposition of A. It can be shown that n x n matrices P1, P2, ... , Pk are uniquely determined by the following three

conditions:

(i)

(ii)P,P1 = O if j 36 t,

(iii)S = A11P1 + A2P2 + ... + AkPk.

Proof.

Note that

k

First, derive that P, S = SP; = \j P,. Then, this implies that .X P1 = >ahPj P1. (A).

h=1

Hence, \jPjPA(A) = \h PiPh(A). Thus, PiPh(A) = 0 whenever j h. Therefore, it follows that P1 = P1(A) = P2P}(A).

Observation IV-1-15. Let A = S + N be the S-N decomposition of an n x n matrix A. Let T be an n x n invertible matrix such that if we set A = T-1ST, then A = diag[A1I1, A2I2, ... , AkIk], Where A1, A2, ... , Ak are distinct eigenvalues of S (and also of A), I, is the m3 x mj identity matrix, and m3 is the multiplicity

of the eigenvalue A,. It is easy to show that

(i) if we set M = T-'NT, then M is nilpotent, MA = AM, and M =

diag[M1i M2,... , Mk}, where Mj are mj x m j nilpotent matrices,

. S = A1P1 + A2P2 + ... + AkPk.

Therefore, P. = P, (S) = P? (A) (j = 1, 2, ... , k) (cf. Observation IV-1-14).

The following two remarks concern real diagonalizable matrices.

Remark IV-1-16. Let A be a real nxn diagonalizable matrix and let A1, A2 ... , An be the eigenvalues of A. Then, there exists a real n x n invertible matrix P such that

(1)in the case when all eigenvalues A j (j = 1,2,... , n) are real, then P-1 AP is a real diagonal matrix whose entries on the main diagonal are A1, A2, ... , An,

(2)in the case when all eigenvalues are not real, then n is an even integer 2m and P-1A fP = diag[D1, D2, ... , DmJ, where A23_1 = a, + ibl, A2J = a) - ibj, and

Dj

abj a,

(3) in other cases, P1AP = diag[D1, D2i ... , Dhj, where A23_1 = a)+ib,, A2.1 =

b, for j = 1, 2, ... , h - 1 and Dh is a real diagonal

a j

matrix whose entries on the main diagonal are A, (j = 2h - 1,... , n).

Remark IV-1-17. For any given real n x n matrix A, there exists a sequence

{Bk : k = 1, 2,. ..} of real n x n diagonalizable matrices such that lim Bk = A.

This can be proved in the following way:

(i)let A = S + N be S-N decomposition of A,

(ii)using Remark IV-1-16, assume that S = diag[D1, D2, ... , Dhj, as in (3) of

Remark IV-1-16,

(iii)find the form of N by SN = NS,

(iv)triangularize N without changing S,

(v)use a method similar to the proof of Lemma IV-1-4.

Details of proofs of Remark IV-1-16 and IV-1-17 are left to the reader as exercises. Now, we give two examples of calculation of the S-N decomposition.

20 -16 2

N=A - S= 10 20 -16 2

-40 32 -4