In mathematics, the Hadamard product (also known as the element-wise product, entrywise product: ch. 5 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.
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:: 479
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:
Furthermore, a Hadamard matrix-vector product can be expressed as:
where is the vector formed from the diagonals of matrix M.
The Hadamard product of two positive-semidefinite matrices is positive-semidefinite. This is known as the Schur product theorem, after Russian mathematician Issai Schur. For two positive-semidefinite matrices A and B, it is also known that the determinant of their Hadamard product is greater than or equal to the product of their respective determinants:
In programming languages
Hadamard multiplication is built into certain programming languages under various names. In MATLAB, GNU Octave, GAUSS and HP Prime, it is known as array multiplication, or in Juliabroadcast multiplication, with the symbol .*. In Fortran, R,APL, J and Wolfram Language (Mathematica), it is done through simple multiplication operator * or ×, whereas the matrix product is done through the function matmul, %*%, +.×, +/ .* and the . operators, respectively.
In Python with the NumPy numerical library, multiplication of array objects as a*b produces the Hadamard product, and multiplication as a@b produces the matrix product. With the SymPy symbolic library, multiplication of array objects as both a*b and a@b will produce the matrix product, the Hadamard product can be obtained with a.multiply_elementwise(b).
In C++, the Eigen library provides a cwiseProduct member function for the Matrix class (a.cwiseProduct(b)), while the Armadillo library uses the operator % to make compact expressions (a % b; a * b is a matrix product). R package matrixcalc introduces function hadamard.prod() for Hadamard Product of numeric matrices or vectors.
The Hadamard product appears in lossy compression algorithms such as JPEG. The decoding step involves an entry-for-entry product, in other words the Hadamard product.
In image processing, the Hadamard operator can be used for enhancing, suppressing or masking image regions. One matrix represents the original image, the other acts as weight or masking matrix.
It is also used to study the statistical properties of random vectors and matrices.
Other Hadamard operations are also seen in the mathematical literature, namely the Hadamard root and Hadamard power (which are in effect the same thing because of fractional indices), defined for a matrix such that:
^Ha D., Dai A.M., Le Q.V. (2017). "HyperNetworks". The International Conference on Learning Representations (ICLR) 2017. – Toulon, 2017.: Page 6. arXiv:1609.09106.((cite journal)): CS1 maint: multiple names: authors list (link)