Linear Algebra, Theory And Applications, 2012a

98 DETERMINANTS
Corollary 3.4.3 Let A i and B i be n × n matrices and suppose
A 0 + A 1 λ + ···+ A m λ m = B 0 + B 1 λ + ···+ B m λ m
for all |λ| large enough. Then A i = B i for all i. Consequently if λ is replaced by any n × n
matrix, the two sides will be equal. That is, for C any n × n matrix,
A 0 + A 1 C + ···+ A m C m = B 0 + B 1 C + ···+ B m C m .
Proof: Subtract and use the result of the lemma.
With this preparation, here is a relatively easy proof of the Cayley Hamilton theorem.
Theorem 3.4.4 Let A be an n×n matrix and let p (λ) ≡ det (λI − A) be the characteristic
polynomial. Then p (A) =0.
Proof: Let C (λ) equal the transpose of the cofactor matrix of (λI − A) for|λ| large.
(If |λ| is large enough, then λ cannot be in the finite list of eigenvalues of A and so for such
λ, (λI − A) −1 exists.) Therefore, by Theorem 3.3.18
C (λ) =p (λ)(λI − A) −1 .
Note that each entry in C (λ) is a polynomial in λ havingdegreenomorethann − 1.
Therefore, collecting the terms,
C (λ) =C 0 + C 1 λ + ···+ C n−1 λ n−1
for C j some n × n matrix. It follows that for all |λ| large enough,
(λI − A) ( C 0 + C 1 λ + ···+ C n−1 λ n−1) = p (λ) I
and so Corollary 3.4.3 may be used. It follows the matrix coefficients corresponding to equal
powers of λ are equal on both sides of this equation. Therefore, if λ is replaced with A, the
two sides will be equal. Thus
0=(A − A) ( C 0 + C 1 A + ···+ C n−1 A n−1) = p (A) I = p (A) .
3.5 Block Multiplication Of Matrices
Consider the following problem
)
C
(
A B
D
)(
E F
G
H
You know how to do this. You get
(
AE + BG AF + BH
CE + DG CF + DH
Now what if instead of numbers, the entries, A, B, C, D, E, F, G are matrices of a size such
that the multiplications and additions needed in the above formula all make sense. Would
the formula be true in this case? I will show below that this is true.
Suppose A is a matrix of the form
⎛
A =
⎜
⎝
)
.
A 11 ··· A 1m
⎞
.
.
. ..
.
.
⎟
⎠ (3.17)