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Recap of lecture on 4/6
Key steps in showing any real symmetric matrix
A
has a basis of
eigenvectors
(1)
The eigenvalues of a real symmetric matrix are real.
We will not prove this now.
(2)
Invariant subspaces
Recall that
W
is
invariant
for
A
provided
A
(
W
)
⊂
W
. That is ,
Aw
is in
W
whenever
w
is in
W
.
Fact
W
⊥
is also invariant , because
A
is symmetric.
Proof
W
⊥
consists of vectors
u
such that
u
•
w
= 0 for all
w
in
W
. We must
show that
Au
•
w
= 0 for all
w
in
W
. We have
Au
•
w
=
u
•
A
T
w
=
u
•
Aw
= 0
because
Aw
is in
W
.
(3) Theorem
If
W
is an invariant subspace for the symmetric
n
×
n
matrix
A
, then there is an eigenvector of
A
lying in
W
.
The proof is in several steps.
(a)
Choose an orthonormal basis
{
u
1
,...,u
p
}
for
W
and choose an orthonor
mal basis
{
u
p
+1
,...,u
n
}
for
W
⊥
. Then
α
=
{
u
1
,...,u
n
}
is an orthonormal
basis
R
n
. Let
±
denote the standard basis for
R
n
. Let
P
=
P
±
α
be the
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View Full Documentorthogonal matrix
P
= [
u
1
,...,u
n
] and recall that
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 Spring '07
 Hutchings
 Linear Algebra, Algebra, Eigenvectors, Vectors

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