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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

I D

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

1

0

1

0

1

0

1

1

1

1

1

1

1

1

0

1

0

1

1

0

1

0

0 i

0

− i

⎤ ⎡

⎥ ⎢

1

1

0

0

1 i i

2 i

3

0

1

0

1

1 i

1

− i i

1

2 i

4 i

6

1

0

1

0

1

1

1

1 i

1

3 i

6 i

9

0 i

0

− i

1

− i

1

⎤ i

⎤ ⎡

0

0

1

1

= F

4

. �

0

0

1

1

1

1

0

0

1

− 1

0

0

0

0

1

1

0

0

1

− 1

0

0

1

1

⎤ ⎡

⎥ ⎢

0

0

1

0

1

0

0

0

0

0

0

1

0

0

0

1

2

MIT OpenCourseWare http://ocw.mit.edu

18.06SC Linear Algebra

Spring 2011

For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .

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