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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.
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; a product involved in the Hadamard factorization theorem for entire ...
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 .
Frobenius inner product, the dot product of matrices considered as vectors, or, equivalently the sum of the entries of the Hadamard product; Hadamard product of two matrices of the same size, resulting in a matrix of the same size, which is the product entry-by-entry; Kronecker product or tensor product, the generalization to any size of the ...
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,
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.
While the order of a Hadamard matrix must be 1, 2, or a multiple of 4, regular Hadamard matrices carry the further restriction that the order must be a square number. The excess, denoted E(H ), of a Hadamard matrix H of order n is defined to be the sum of the entries of H. The excess satisfies the bound |E(H )| ≤ n 3/2.
The article (both title and content) suggests that the name "Hadamard product" is only used for matrices. However, a similar element-wise/entrywise product could be applied to vectors, or to multilinear tensors (multidimensional arrays), not just to the specific case of matrices (two-dimensional arrays).