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.