Math 417 homework 10
Solutions
Problem 1
(a) Show: if
v
1
, v
2
are eigenvectors of a matrix
A
for the same eigenvalue
λ
,
then their linear combinations
α
1
v
1
+
α
2
v
2
are also eigenvectors (except for
those = 0) for that eigenvalue
λ
.
Answer:
A
(
α
1
v
1
+
α
2
v
2
) =
α
1
Av
1
+
α
2
Av
2
=
λα
1
v
1
+
λα
2
v
2
=
λ
(
α
1
v
1
+
α
2
v
2
)
.
(b) Consider
A
=
SDS

1
where
S
=
1
0
2
0

1
0
2
0
0
,
D
=

1
0
0
0
2
0
0
0

1
.
(1) What are the eigenvalues of
A
? What are the eigenvectors? [There is a hard
way and an easy way to do this.]
Answer: the eigenvalues are the diagonal entries of
D
;

1 is a double root and
2 is single. The eigenvectors are the columns of
S
. Column
i
is an eigenvector
for the eigenvalue in column
i
row
i
of
D
.
(2) Find an
orthogonal diagonalization
of
A
, i.e. an orthogonal 3
×
3 matrix
U
with
A
=
UDU
T
.
Answer:
we already have a diagonalization, but
S
is not an orthogonal ma
trix.
But part (a) tells us that the GramSchmidt algorithm can be used to
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 Fall '07
 ELLING
 Linear Algebra, Eigenvectors, Vectors, Matrices, ax, GramSchmidt algorithm, orthogonal diagonalization

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