Walsh matrix

The Walsh matrices are a special case of Hadamard matrices where the rows are rearranged so that the number of sign changes in a row is in increasing order.

are given by the recursive formula (the lowest order of Hadamard matrix is 2): and in general for 2 ≤ k ∈ N, where ⊗ denotes the Kronecker product.

Finally, we re-order the rows of the matrix according to the number of sign changes in ascending order.

where the successive rows have 0, 3, 1, and 2 sign changes (we count the number of times we switch from a positive 1 to a negative 1, and vice versa).

The sequency ordering of the rows of the Walsh matrix can be derived from the ordering of the Hadamard matrix by first applying the bit-reversal permutation and then the Gray-code permutation:[2] where the successive rows have 0, 1, 2, 3, 4, 5, 6, and 7 sign changes.

Hadamard matrix of order 16 multiplied with a vector
Naturally ordered Hadamard matrix permuted into sequency-ordered Walsh matrix. The number of sign changes per row in the naturally ordered matrix is (0, 15, 7, 8, 3, 12, 4, 11, 1, 14, 6, 9, 2, 13, 5, 10), in the sequency-ordered matrix the number of sign changes is consecutive.
LDU decomposition of a Hadamard matrix. The ones in the triangular matrices form Sierpinski triangles . The entries of the diagonal matrix are values from Gould's sequence , with the minus signs distributed like the ones in Thue–Morse sequence .