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Quiz VII
1. The matrix
A
=
⎛
⎝
201
010
102
⎞
⎠
is symmetric. Find a basis for
R
3
consisting of
eigenvectors for
A
.
Solution:
Since
A
is symmetric,
there will be
an
orthonormal
basis consisting of
eigenvectors of
A
, but you do not need to use that to solve the problem. First ±nd
the eigenvalues
det
⎛
⎝
2
−
λ
01
−
λ
0
10
2
−
λ
⎞
⎠
=(2
−
λ
)
2
(1
−
λ
)
−
(1
−
λ
)=(1
−
λ
)(3
−
4
λ
+
λ
2
)
.
So the eigenvalues are 3 and 1. Solving for the eigenspaces,
±v
1
=
⎛
⎝
1
0
1
⎞
⎠
is an
eigenvector for the eigenvalue 3. For the eigenvalue 1,
A
−
I
=
⎛
⎝
101
000
⎞
⎠
,
and the reduced row echelon form of that is just
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This note was uploaded on 04/12/2011 for the course MATH 33a taught by Professor Lee during the Fall '08 term at UCLA.
 Fall '08
 lee
 Eigenvectors, Vectors

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