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Handout on the eigenvectors of distinct eigenvalues
9/30/04
This handout shows,
f
rst, that eigenvectors associated with distinct
eigenvalues of an abitrary square matrix are linearly indpenent, and sec-
ond, that all eigenvectors of a symmetric matrix are mutually orthogonal.
First we show that all eigenvectors associated with distinct eigenval-
ues of an abitrary square matrix are mutually linearly independent:
Suppose
k
(
k
≤
n
)
eigenvalues
{
λ
1
, ..., λ
k
}
of
A
are distinct,
and take any corresponding eigenvectors
{
v
1
, ..., v
k
}
,
de
f
ned by
v
j
6
=0
,Av
j
=
λ
j
v
j
for
j
=1
, ..., k.
Then,
{
v
1
, ..., v
k
}
are linearly
independent.
First consider two such eigenvectors. Suppose we have eigenvalue
λ
with eigenvector
v
, and eigenvalue
µ
with eigenvector
w
,
λ
6
=
µ
.W
e
will show that
αv
+
βw
=0
⇒
α
=
β
=0
, implying that
v
and
w
are
linearly independent.
So suppose we have

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