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Winter 2011
•
Math 67
•
Linear Algebra
Writing Assignment 3
Exercise 1.
The circulant matrix
X
=
x
1
x
2
x
3
x
4
x
4
x
1
x
2
x
3
x
3
x
4
x
1
x
2
x
2
x
3
x
4
x
1
∈
C
4
×
4
has an eigenvector (1
,ζ,ζ
2
,ζ
3
), where
ζ
is a 4th root of unity. The associated eigen
value is
x
1
+
ζx
2
+
ζ
2
x
3
+
ζ
3
x
4
.
Proof.
Observe that the following system of equations is consistent for
λ
=
x
1
+
ζx
2
+
ζ
2
x
3
+
ζ
3
x
4
, given that
ζ
4
= 1.
x
1
x
2
x
3
x
4
x
4
x
1
x
2
x
3
x
3
x
4
x
1
x
2
x
2
x
3
x
4
x
1
1
ζ
ζ
2
ζ
3
=
x
1
+
ζx
2
+
ζ
2
x
3
+
ζ
3
x
4
x
4
+
ζx
1
+
ζ
2
x
2
+
ζ
3
x
3
x
3
+
ζx
4
+
ζ
2
x
1
+
ζ
3
x
2
x
2
+
ζx
3
+
ζ
2
x
4
+
ζ
3
x
1
=
λ
λζ
λζ
2
λζ
3
±
Exercise 2.
The matrix
Z
=
1 1 0 0
0 1 1 0
0 0 1 1
1 0 0 1
∈
R
4
×
4
has eigenvalues 0,1

i
,1 +
i
, and 2. The eigenvectors of a
general
4
×
4 circulant
matrix are linearly independent.
Proof.
Since
Z
is a circulant matrix, it follows from the result of problem 1 that it
has an eigenvector (1
,ζ,ζ
2
,ζ
3
) with associated eigenvalue 1+
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This note was uploaded on 11/13/2011 for the course MATH 67 taught by Professor Schilling during the Winter '07 term at UC Davis.
 Winter '07
 Schilling
 Linear Algebra, Algebra

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