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