Principles of Mathematics in Operations Research

4.3 Diagonal Form of a Matrix 59

Proposition 4.3.10 m(s) = ITf=1(s - Xi)mi, then

C" = Af[(A - Ai)mi] e • • -®AT[(A - Xk)mk],

where © is the direct sum of vector spaces.

Theorem 4.3.11 d(s) = n^=1(s - A*)"*, m(s) = n*=1{s - A*)mi.

i. dim(Af[{A - A*)mi]) = m;

ii. If columns of n x ni matrices Bi form bases for Af[(A — Aj)m*] and B =

[Bi\ • • • |.Bfc], then B is nonsingular and

B'XAB

M

Ak

where Ai are ni x n,;

Hi. Independent of the bases chosen for Af[(A — A;)mi],

det(sl - A\) = (a - Xi)"*;

iv. Minimal polynomial of Ai is (s — Xi)mi.

Example 4.3.12

"0 10'

0 01

2-5 4

A = , d(s) =

Ai = 1, m = 2; A2 = 2, n 2 = 1.

s-1 0

0 s -1

-2 5s-4

= (s-l)^2 (s-2)=0.

[A-- AiJ] =

"-1 10"

0 -1 1

2-5 3

, dim{Af[{A - A:)]) = 1< 2 = m(!)

mi > 1 => mi = 2 => m(s) = (s - l)^2 (s - 2) = d(s).

[A--Ai/]

(^2) =
"1 -2 1"
2-4 2
4-8 4
, dim(M[(A - Aj)^2 ]) = 2
« 11 = (l,0,-l)r) «i2 = (0,l,2)T, B1 =
A 2 = 2, [;4 - A 2 7] =
10
01
-12
"-2
0
2
10"
-2 1
-5 2
dim(N[{A - Aa)]) = 1.