Download Homework 5 - UMass Math

Math 545
Homework # 5: due Mar 22, 2017
1. For an n × n Fourier matrix F :
(a) calculate F F , and use this to find F −1 ;
(b) find a permutation of the columns of F that produces F , so that F P = F ;
(c) use these to find F 2 and F 4 .
2. Let w be a primitive sixth root of unity, w6 = 1. Then w2 is a cube root. Factorize the
Fourier matrix F6 into the block form
I D
F3 0
F6 =
P,
I −D
0 F3
where P is a permutation matrix; express D and F3 using powers of w.
3. Compute y = F8 c by the three steps of the Fast Fourier Transform, for each of c =
(1, 0, 1, 0, 1, 0, 1, 0) and c = (0, 1, 0, 1, 0, 1, 0, 1).
4. For any n, find the determinant of the permutation matrix Pn with 1s on the reverse
diagonal:


0 0 0 1
 0 0 1 0 

P4 = 
 0 1 0 0 .
1 0 0 0
5. Calculate the determinant of the 4 × 4 Vandermonde

1 a a 2 a3
 1 b b2 b3
V4 = 
 1 c c2 c3
1 d d2 d3
matrix,


.
