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

1. Walsh matrices are almost idempotent:
   W(2^k) * W(2^k) = 2^k * I
2. So:
   V * H(2^k) * H(2^k) = 2^k * V
3. Hadarmard matrices satisfy:
   H(n) * H(n)^t = n * I
   W(2^k)^t = W(2^k)
   
4. Uniqueness - up to the following transforms:
1. interchanging rows or columns
2. multiplying a row or column by -1
5. Uniqueness
1. 1, 2, 4, 8, 12 are unique
2. 16 has 5 matrices
3. 20 has 3
4. 60 has 24
5. 28 has 487
6. 32, 36, 40 have millions