Principles of Mathematics in Operations Research

4.3 Diagonal Form of a Matrix 57

4.3.2 Repeated Eigen Values with Full Kernels

In this case, (recall that d(s) = Yii=i(s ~ A;)ni)i we have dimAf([A — Xil]) —

Tii,Vz. Thus, there exists n* linearly independent vectors in Af ([A — Xil]), each

of which is an eigen vector associated with A;, Vi.

Ai •H- Vn,Vl2,...,Vini

A 2 f* V 2 l,V22,---,V2n 2

Afc *-t Vkl,Vk2,...,Vknk

Ur=i {v»i}?ii 1S linearly independent (Exercise). Thus, we have obtained

n linearly independent vectors, which constitute a basis. Consequently, we get

S~^1 AS =

Ai

Example 4.3.5

A =

3 1 -1

13-1

00 2

d(s) = det(sl - A) =

s-3 -1 1

-1 s-3 1

0 0 s-2

= 0

= (* - 3)^2 (s - 2) - (a - 2) = (a - 2)[(* - 3)^2 - 1]

= (* - 2)(s - 4)(a - 2) = (a - 2)^2 (s - 4).

=> Ai = 2, ni = 2 and A2 = 4, 712 = 1.

A-\iI =

1 1 -1

1 1 -1

00 0

dim(Af([A - Aj/])) = 2.

vii = (l)-l,0)r,«i2 = (0,l,l)T.

-1 1 -1

A-\ 2 I= 1-1-1

0 0-2