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Math 2940
Worksheet: Ch. 7 Eigenvalues and Eigenvectors
November 12, 2009
1. Determine if the following matrix is diagonalizable. If possible, ﬁnd an invertible
S
and a
diagonal
D
such that
S

1
AS
=
D
.
A
=
2 0 1
0 1 0
0 0 1
2. For which values of constants
a
,
b
, and
c
are the following matrices diagonalizable?
(a)
1
a b
0 2
c
0 0 3
(b)
0 0 0
1 0
a
0 1 0
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View Full DocumentTo Diagonalize an
n
×
n
Matrix,
A
. . .
1. Find all eigenvalues with their algebraic multiplicities. To do this, solve the equation
det(
A

λI
) = 0 for
λ
.
2. For each eigenvalue,
λ
, ﬁnd the corresponding eigenspace,
E
λ
= ker(
A

λI
). The
dimension of
E
λ
is the geometric multiplicity of
λ
.
3. If the geometric multiplicity of each eigenvalue is the same as its algebraic multiplic
ity, or equivalently, if the geometric multiplicities add up to
n
, then the matrix is
diagonalizable.
4. If the matrix is diagonalizable, collect together the bases of each eigenspace. The
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 Spring '06
 PANTANO
 Eigenvectors, Multivariable Calculus, Vectors

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