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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 .
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 mathematics, a complex Hadamard matrix H of size N with all its columns (rows) mutually orthogonal, belongs to the Butson-type H(q, N) if all its elements are powers of q-th root of unity, =, =,, …,.
A number of methods for constructing regular Hadamard matrices are known, and some exhaustive computer searches have been done for regular Hadamard matrices with specified symmetry groups, but it is not known whether every even perfect square is the order of a regular Hadamard matrix. Bush-type Hadamard matrices are regular Hadamard matrices of ...
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, = [], Hadamard matrices of every permissible size up to 100 except for 92 are produced.
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.
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}} ) .
In words, when given an x, it is not possible to find another x' such that the hashing function would create a collision. A hash function has strong collision resistance when, given a hashing function H, no arbitrary x and x' can be found where H(x)=H(x'). In words, no two x's can be found where the hashing function would create a collision.