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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.
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}} ) .
Let H be a Hadamard matrix of order n.The transpose of H is closely related to its inverse.In fact: = where I n is the n × n identity matrix and H T is the transpose of H.To see that this is true, notice that the rows of H are all orthogonal vectors over the field of real numbers and each have length .
Hadamard's maximal determinant problem, named after Jacques Hadamard, asks for the largest determinant of a matrix with elements equal to 1 or −1. The analogous question for matrices with elements equal to 0 or 1 is equivalent since, as will be shown below, the maximal determinant of a {1,−1} matrix of size n is 2 n−1 times the maximal determinant of a {0,1} matrix of size n−1.
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. For N = 2 , 3 {\displaystyle N=2,3} and 5 {\displaystyle 5} all complex Hadamard matrices are equivalent to the Fourier matrix F N {\displaystyle F_{N}} .
Hadamard matrices of every permissible size up to 100 except for 92 are produced. In his 1933 paper, Paley says “It seems probable that, whenever m is divisible by 4, it is possible to construct an orthogonal matrix of order m composed of ±1, but the general theorem has every appearance of difficulty.”
The Hadamard transform can also be combined with a wide variety of methods to accommodate among-sites rate heterogeneity, [11] using continuous distributions rather than the discrete approximations typically used in maximum likelihood phylogenetics [12] (although one must sacrifice the invertibility of the Hadamard transform to use certain ...
For example, the fast Walsh–Hadamard transform (FWHT) may be used in the analysis of digital quasi-Monte Carlo methods. In radio astronomy, Walsh functions can help reduce the effects of electrical crosstalk between antenna signals.