Exercises on complex matrices; fast Fourier transform
Problem 26.1: Compute the matrix F
2
.
Solution: F
2
=
�
1 1
1 w
�
, where w = e i 2
π
/2 = − 1. Hence
F
2
=
� 1 1
1 − 1
�
.
Problem 26.2: Find the matrices D and P used in the factorization:
� � � �
F
4
=
I D
I − D
F
2
F
2
P
Hint: D is created using fourth roots, not square roots, of 1. Check your answer by multiplying.
Solution: We computed F
2
=
�
1 1
1 − 1
� in the previous problem.
D =
�
1 0
0 i
� contains the first two fourth roots of 1 (and others, -1 and i ).
− D contains the
P is a permutation matrix that arranges the components of the incom­ ing vector so that its even components come first. For F
4
, that means swap­ ping the first and second components:
P
⎡
⎢
⎢
⎣ x
0 x
1 x
2 x
3
⎤
⎥
⎥
⎦
⎡
=
⎢
⎢
⎣ x
0 x
2 x
1 x
3
⎤
⎥
⎥
⎦
.
So, P =
⎡ 1 0 0 0 ⎤
⎢
⎢
⎣
0 0 1 0
0 1 0 0
⎥
⎥
⎦
0 0 0 1
.
1

Finally, we check our work by multiplying:
�
I
I
D
− D
� �
F
2
F
2
�
P =
=
=
=
⎡
⎢
⎢
⎣
1 0
0 1
1 0
0 1
1
0
− 1
0
0 i
0
− i
⎤ ⎡
⎥ ⎢
⎥
⎦
⎢
⎣
1
1
0
0
⎡
⎢
⎢
⎣
⎡
⎢
⎢
⎣
⎡
⎢
⎢
⎣
1 0
0 1
1
0
0 i
⎤ ⎡
⎥ ⎢
1 0
1 0
1 0
0 1
− 1
0
0
− i
⎥
⎦
⎢
⎣
⎤ 1 1 1 1
1
1
1 i
− 1
− i
− 1 − i
1 − 1
− 1
⎤ i
1 1 1 1
1 i i 2 i 3
1 i 2
1 i 3 i i 4
6 i 6 i 9
⎥
⎥
⎦
=
⎥
⎥
⎦
F
4
.
0 1
0 1
�
1 0
− 1 0 0
0 1
0 1
1
− 1
0
0
0
1
− 1
0
0
1
− 1
⎥
⎥
⎦
⎤ ⎡
⎥ ⎢
⎥
⎦
⎢
⎣
1 0 0 0
0 0 1 0
0 1 0 0
0 0 0 1
⎤
⎤
⎥
⎥
⎦
2

MIT OpenCourseWare http://ocw.mit.edu
18.06SC Linear Algebra
Spring 2011
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