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A Hadamard matrix H is skew if + =. A skew Hadamard matrix remains a skew Hadamard matrix after multiplication of any row and its corresponding column by −1. This makes it possible, for example, to normalize a skew Hadamard matrix so that all elements in the first row equal 1.
The Hadamard product operates on identically shaped matrices and produces a third matrix of the same dimensions. In mathematics, the Hadamard product (also known as the element-wise product, entrywise product [1]: ch. 5 or Schur product [2]) is a binary operation that takes in two matrices of the same dimensions and returns a matrix of the multiplied corresponding elements.
Two complex Hadamard matrices are called equivalent, written , if there exist diagonal unitary matrices , and permutation matrices, such that =. Any complex Hadamard matrix is equivalent to a dephased Hadamard matrix, in which all elements in the first row and first column are equal to unity.
The Hadamard transform H m is a 2 m × 2 m matrix, the Hadamard matrix (scaled by a normalization factor), that transforms 2 m real numbers x n into 2 m real numbers X k.The Hadamard transform can be defined in two ways: recursively, or by using the binary (base-2) representation of the indices n and k.
The size of a Hadamard matrix must be 1, 2, or a multiple of 4. The Kronecker product of two Hadamard matrices of sizes m and n is an Hadamard matrix of size mn. By forming Kronecker products of matrices from the Paley construction and the 2 × 2 matrix, = [],
The Hadamard code is a linear code, and all linear codes can be generated by a generator matrix.This is a matrix such that () = holds for all {,}, where the message is viewed as a row vector and the vector-matrix product is understood in the vector space over the finite field.
For example, for the 2×2 matrix ... Vectorization is an algebra homomorphism from the space of n × n matrices with the Hadamard ... In Matlab/GNU Octave a matrix A ...
In computational mathematics, the Hadamard ordered fast Walsh–Hadamard transform (FWHT h) is an efficient algorithm to compute the Walsh–Hadamard transform (WHT). A naive implementation of the WHT of order n = 2 m {\displaystyle n=2^{m}} would have a computational complexity of O( n 2 {\displaystyle n^{2}} ) .