Principles of Mathematics in Operations Research

54 4 Eigen Values and Vectors

4.2 Eigen Values and Eigen Vectors

Definition 4.2.1 The number X is an eigen value of A, with a corresponding

nonzero eigen vector v such that Av = Xv.

The last equation can be organized as (XI—A)v = 0. In order to have a non-

trivial solution v 7^ 9, the corresponding null space (kernel) M(XI — A) should

contain vectors other than 9. Thus, the kernel has dimension larger than 0,

which means we get at least one zero row in Gaussian elimination. Therefore,

(XI — A) is singular. Hence, A should be chosen such that det(AJ — A) = 0.

This equation is known as characteristic equation for A.

d(s) = det(sl - A) = s^11 + dxsn+1 + • • • + dn = 0.

Then, the eigen values are the roots.

k

d(s) = (s- X,r (s - A 2 )"^2 ...(«- Xk)n- = Y[(s - K)ni-

The sum of multiplicities should be equal to the dimension, i.e. Y^i ni — n-

The sum of n-eigen values equals the sum of n-diagonal entries of A.

Ai + • • • + An = niAi + • • • + nfcAfc = ou + • • • + ann.

This sum is known as trace of A. Furthermore, the product of the n-eigen

values equals the determinant of A.

n*<=n

A

"'

=detA

i=l

Remark 4.2.2 If A is triangular, the eigen values X\,.

entries an,... ,ann.

., Xn are the diagonal

Example 4.2.3

A =

I 0 0

II o

u 4 4

det.4 = i(l)| = |(property 8).

d(s) =

8~\ 0

-i 8-1

0

0

s —

= 1-2 (*-l)

So, Ai = \ = an, A 2 = 1 = a 2 i, A 3 = | = a 33. Finally,

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