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 *F*^{n} 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 *F*^{n} . Let

and

Define
*B* = *x*^{-1} *A x* , then B is block upper triangular

Furthermore
*A x*_{1} = *x*_{1} *B*

(
, *xv* =
=
)

Note :
1.
If
*B*_{21} = 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* = *v*_{1} + *i v*_{2} is the Eigenvector of A
is also the Eigenvector of A