Kenneth Kuttler

13.2. FUNDAMENTAL THEORY AND GENERALIZATIONS 303

and so ∑s BisAs j =

∑s

(0 Iri×ri 0

 0Ips×ps

( 0 Ips×ps 0)

 0Iq j×q j



0 Iri×ri 0)

B∑s

 0Ips×ps

( 0 Ips×ps 0)

 0Iq j×q j



0 Iri×ri 0)

BIA

 0Iq j×q j

=(

0 Iri×ri 0)

 0Iq j×q j

which equals the i jth block of BA. Hence the i jth block of BA equals the formal multipli-cation according to matrix multiplication, ∑s BisAs j. ■

Example 13.2.3 Let an n×n matrix have the form

A =

(a bc P

)

where P is n−1×n−1. Multiply it by

B =

(p qr Q

)

where B is also an n×n matrix and Q is n−1×n−1.

You use block multiplication(a bc P

)(p qr Q

(ap+br aq+bQpc+Pr cq+PQ

)

Note that this all makes sense. For example, b = 1× n− 1 and r = n− 1× 1 so br is a1×1. Similar considerations apply to the other blocks.

Here is an interesting and significant application of block multiplication. In this the-orem, pM (t) denotes the characteristic polynomial, det(tI−M) . Thus the zeros of thispolynomial are the eigenvalues of the matrix M.

Theorem 13.2.4 Let A be an m×n matrix and let B be an n×m matrix for m≤ n. Then

pBA (t) = tn−m pAB (t) ,

so the eigenvalues of BA and AB are the same including multiplicities except that BA hasn−m extra zero eigenvalues.

13.2. FUNDAMENTAL THEORY AND GENERALIZATIONS 303and so )', BisAs; =0 0YE (0 den OBL Incr [(O tnx 0 JAY lyynay; 0 00 0=(0 den O)BYY Inxre | (0 Inne 0 JAY typxaiy\ 0 00 0=(0 dyn 0 )BIAT Tyg |= (0 Ixy 0) BAL Iya,0 0which equals the ij” block of BA. Hence the ij” block of BA equals the formal multipli-cation according to matrix multiplication, )°, BjsAs;.Example 13.2.3 Let an n x n matrix have the form(oo)where Pisn—1xn—1. Multiply it bya-(! ‘)r Qwhere B is also ann Xn matrix and Q isn—1xn-—1.You use block multiplicationa b Pp q \_{ apt+br aq+bQc P r Q 7 pe+Pr cq+PQNote that this all makes sense. For example, b = 1 xn—1 andr=n—1x1sobrisa1 x 1. Similar considerations apply to the other blocks.Here is an interesting and significant application of block multiplication. In this the-orem, py (t) denotes the characteristic polynomial, det (t!—_M). Thus the zeros of thispolynomial are the eigenvalues of the matrix M.Theorem 13.2.4 Let A be anm xn matrix and let B be ann x m matrix for m <n. Thenn—mtpea (t) = PaB(t),so the eigenvalues of BA and AB are the same including multiplicities except that BA hasn—m extra zero eigenvalues.