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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

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