Principles of Mathematics in Operations Research

4 Eigen Values and Vectors

10 r

111

010

, 5-

V2

-1 _

= (l,l,0)r

• I I I •
  2 2 2
  0 0 1
  1 1 1
  .2 2 2.

, S~^1 AS =

"200"

020

004

s =

4.3.3 Block Diagonal Form

In this case, we have

3i 9 iii > 1, dim(Af[A - A;/]) < n.

Definition 4.3.6 The least degree monic (the polynomial with leading co-

efficient one) polynomial m(s) that satisfies m(A)=0 is called the minimal

polynomial of A.

Proposition 4.3.7 The following are correct for the minimal polynomial.

i. m(s) divides d(s);

ii. m(Xi) =0, Vi = 1,2,..., k;

Hi. m(s) is unique.

Example 4.3.8

A =

c 1 0

OcO

00c

, d{s) = det(s7 - A)

s-c -1 0

0 s-c 0

0 0 s-c

= (s- c)^3 = 0.

Ax = c, ni = 3. m(s) =? (s — c), (s — c)^2 , (s — c)^3

[A - Xil] =

010

000

000

[A - Ai/]^2 =

0 10

000

000

010

000

000

^0 3 => m(s) ^(s- c).

= 0 3 => m(s) - (s - c)^2.

Then, to find the eigen vectors

(A - d)x = <?<£>

010

000

000

x = 6 =>• Dn

"1"

0

0

, V\2 =

"0"

0

1

Proposition 4.3.9

d(s) = nU(* - ^)"% m(s) = n?=1(a - Xi)m\ 1 < rrn < m, i = 1,2,.

*W - Ai/)] § MU - V)

2

] i • • • i MU - vn

= jVp - A*/)™^^1 ] = • • • = N{{A - Xil)ni]

,k.