Walsh matrix

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.
Binary Hadamard matrix as a matrix product. The binary matrix (white 0, red 1) is the result with operations in F2. The gray numbers show the result with operations in .
See also: Hadamard matrix

In mathematics, a Walsh matrix is a specific square matrix of dimensions 2n, where n is some particular natural number. The entries of the matrix are either +1 or −1 and its rows as well as columns are orthogonal. The Walsh matrix was proposed by Joseph L. Walsh in 1923.[1] Each row of a Walsh matrix corresponds to a Walsh function.

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. In short, a Hadamard matrix is defined by the recursive formula below and is naturally ordered, whereas a Walsh matrix is sequency-ordered.[1] Confusingly, different sources refer to either matrix as the Walsh matrix.

The Walsh matrix (and Walsh functions) are used in computing the Walsh transform and have applications in the efficient implementation of certain signal processing operations.

  1. ^ a b Kanjilal, P. P. (1995). Adaptive Prediction and Predictive Control. Stevenage: IET. p. 210. ISBN 0-86341-193-2.