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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.
Define the Hadamard canonical factors ():= = / Entire functions of finite order have Hadamard's canonical representation: [1] = = (/) where are those roots of that are not zero (), is the order of the zero of at = (the case = being taken to mean ()), a polynomial (whose degree we shall call ), and is the smallest non-negative integer such that the series = | | + converges.
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.
Example: The Hadamard transform on a 3-qubit register | . Here the amplitude for each measurable state is 1 ⁄ 2 . The probability to observe any state is the square of the absolute value of the measurable states amplitude, which in the above example means that there is one in four that we observe any one of the individual four cases.
This can be applied recursively, as done in the radix-2 FFT and the Fast Walsh–Hadamard transform. Splitting a known matrix into the Kronecker product of two smaller matrices is known as the "nearest Kronecker product" problem, and can be solved exactly [13] by using the SVD. To split a matrix into the Kronecker product of more than two ...
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.