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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.
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 .
In mathematics, the Hadamard product may refer to: Hadamard product of two matrices , the matrix such that each entry is the product of the corresponding entries of the input matrices Hadamard product of two power series , the power series whose coefficients are the product of the corresponding coefficients of the input series
Since the Walsh–Hadamard code is a linear code, the distance is equal to the minimum Hamming weight among all of its non-zero codewords. All non-zero codewords of the Walsh–Hadamard code have a Hamming weight of exactly by the following argument. Let {,} be a non-zero message. Then the following value is exactly equal to the fraction of ...
In mathematics, the Paley construction is a method for constructing Hadamard matrices using finite fields. The construction was described in 1933 by the English mathematician Raymond Paley . The Paley construction uses quadratic residues in a finite field GF( q ) where q is a power of an odd prime number .
Hadamard code and it is the same as the first order Reed–Muller code over the binary alphabet.[4] Normally, Hadamard codes are based on Sylvester's construction of Hadamard matrices, but the term “Hadamard code” is also used to refer to codes constructed from arbitrary Hadamard matrices, which are not necessarily of Sylvester type.
In mathematics, particularly in linear algebra, the Schur product theorem states that the Hadamard product of two positive definite matrices is also a positive definite matrix. The result is named after Issai Schur [ 1 ] (Schur 1911, p. 14, Theorem VII) (note that Schur signed as J. Schur in Journal für die reine und angewandte Mathematik .
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}} .