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Concept
Hadamard product (matrices)
Summary
In mathematics, the Hadamard product (also known as the element-wise product, entrywise product or Schur product) is a binary operation that takes in two matrices of the same dimensions and returns a matrix of the multiplied corresponding elements. This operation can be thought as a "naive matrix multiplication" and is different from the matrix product. It is attributed to, and named after, either French-Jewish mathematician Jacques Hadamard or German-Jewish mathematician Issai Schur.
The Hadamard product is associative and distributive. Unlike the matrix product, it is also commutative.
For two matrices A and B of the same dimension m × n, the Hadamard product (or ) is a matrix of the same dimension as the operands, with elements given by
For matrices of different dimensions (m × n and p × q, where m ≠ p or n ≠ q), the Hadamard product is undefined.
For example, the Hadamard product for two arbitrary 2 × 3 matrices is:
The Hadamard product is commutative (when working with a commutative ring), associative and distributive over addition. That is, if A, B, and C are matrices of the same size, and k is a scalar:
The identity matrix under Hadamard multiplication of two m × n matrices is an m × n matrix where all elements are equal to 1. This is different from the identity matrix under regular matrix multiplication, where only the elements of the main diagonal are equal to 1. Furthermore, a matrix has an inverse under Hadamard multiplication if and only if none of the elements are equal to zero.
For vectors x and y, and corresponding diagonal matrices Dx and Dy with these vectors as their main diagonals, the following identity holds: where x* denotes the conjugate transpose of x. In particular, using vectors of ones, this shows that the sum of all elements in the Hadamard product is the trace of ABT where superscript T denotes the matrix transpose, that is, . A related result for square A and B, is that the row-sums of their Hadamard product are the diagonal elements of ABT: Similarly, Furthermore, a Hadamard matrix-vector product can be expressed as: where is the vector formed from the diagonals of matrix M.
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