MATH 235
Diagonalization  1
Assignment 4a
Not to be handed in.
1. Let
A
be an
n
×
n
matrix. Also, let
α, β, γ
be three distinct eigenvalues of
A
having
corresponding eigenvectors
x
,
y
,
z
, respectively.
Consider the vector
v
=
x
+
y
+
z
.
Can
v
be an eigenvector of
A
corresponding to an eigenvalue
λ
(possibly different from
α, β, γ
? Explain.
2. Recall that the trace of an
n
×
n
matrix
A
= [
a
ij
], denoted by tr(
A
), is the sum of the
diagonal elements, that is, tr(
A
) =
∑
n
i
=1
a
ii
.
(a) Let
C
and
D
be any two
n
×
n
matrices. Prove that tr(
CD
) = tr(
DC
).
(b) Prove that if
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 Fall '08
 CELMIN
 Math, Linear Algebra, Eigenvectors, Vectors, Matrices, Eigenvalue, eigenvector and eigenspace

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