Walsh-Hadamard matrices
Walsh 1:
1
Walsh 2:
1 1
1 -1
Walsh 4:
1 1 1 1
1 -1 1 -1
1 1 -1 -1
1 -1 -1 1
Walsh 8
1 1 1 1 1 1 1 1
1 -1 1 -1 1 -1 1 -1
1 1 -1 -1 1 1 -1 -1
1 -1 -1 1 1 -1 -1 1
1 1 1 1 -1 -1 -1 -1
1 -1 1 -1 -1 1 -1 1
1 1 -1 -1 -1 -1 1 1
1 -1 -1 1 -1 1 1 -1
Notes
- Walsh matrices are almost idempotent:
W(2k) * W(2k) = 2k * I
- So:
V * H(2k) * H(2k) = 2k * V
- Hadarmard matrices satisfy:
H(n) * H(n)t = n * I
W(2k)t = W(2k)
- Uniqueness - up to the following transforms:
- interchanging rows or columns
- multiplying a row or column by -1
- Uniqueness
- 1, 2, 4, 8, 12 are unique
- 16 has 5 matrices
- 20 has 3
- 60 has 24
- 28 has 487
- 32, 36, 40 have millions