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(2k) * W(2k) = 2k * I
  2. So:
    V * H(2k) * H(2k) = 2k * V
  3. Hadarmard matrices satisfy:
    H(n) * H(n)t = n * I
    W(2k)t = W(2k)
  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