Chapter 5 Eigenvalue and Eigenvector II

5.1 Invariance Subspace

F : ( C or ) a field

<Def> A subspace is invariant ( of A )

if , where

Example : 1. If v is Eigenvector of A <v> the eigenspace of A

then <v> is invariant .

u = c u =

2. : the null space of

<Thm 5.5.1>

Let be a subspace of Fn with basis

Thus Let

Then is invariant under

there exist

Recall that

Ax = xB is invariant and

Schur's Thm : , unitary s.t. Q* A Q = T = D + N

Real Shur's Thm : , orthogonal

where or

s.t. if and

<Thm 5.1.2 > Let be invariant under and Let be a basis for

Let be any n - k vectors s.t.

is a basis for Fn . Let

and

Define B = x-1 A x , then B is block upper triangular

Furthermore A x1 = x1 B

( , xv = = )

Note : 1. If B21 = 0 , then is invariant 2. can be a orthogonal matrix

Exercise 5.1.10 : If is invariant under is invariant under A*

Note : If A is Hermitian ( A* = A ) , then

[Real Schur's Thm] or

Note : 1. If is an Eigenvalue of A , then is also an Eigenvalue of A 2. v = v1 + i v2 is the Eigenvector of A is also the Eigenvector of A