In mathematics, the Hadamard product (also known as the elementwise product, entrywise product^{[1]}^{[2]}^{:ch. 5} or Schur product^{[3]}) is a binary operation that takes two matrices of the same dimensions and produces another matrix of the same dimension as the operands, where each element i, j is the product of elements i, j of the original two matrices. It is to be distinguished from the more common matrix product. It is attributed to, and named after, either French mathematician Jacques Hadamard or German mathematician Issai Schur.
The Hadamard product is associative and distributive. Unlike the matrix product, it is also commutative.^{[4]}
Definition
For two matrices A and B of the same dimension m × n, the Hadamard product (or ^{[1]}^{[5]}^{[6]}^{[7]}) is a matrix of the same dimension as the operands, with elements given by^{[4]}
For matrices of different dimensions (m × n and p × q, where m ≠ p or n ≠ q), the Hadamard product is undefined.
Example
For example, the Hadamard product for a 3 × 3 matrix A with a 3 × 3 matrix B is
Properties
 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.^{[8]}

For vectors x and y, and corresponding diagonal matrices D_{x} and D_{y} with these vectors as their main diagonals, the following identity holds:^{[2]}^{:479}
 The Hadamard product is a principal submatrix of the Kronecker product.

The Hadamard product satisfies the rank inequality

If A and B are positivedefinite matrices, then the following inequality involving the Hadamard product is valid:^{[10]}

If D and E are diagonal matrices, then^{[11]}

The Hadamard product of two vectors and is the same as matrix multiplication of one vector by the corresponding diagonal matrix of the other vector:
The mixedproduct property
 , where is Kronecker product
 , where denotes facesplitting product.^{[12]}
 , where is columnwise Khatri–Rao product.
Schur product theorem
The Hadamard product of two positivesemidefinite matrices is positivesemidefinite.^{[4]}^{[9]} This is known as the Schur product theorem,^{[8]} after Russian mathematician Issai Schur. For two positivesemidefinite 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:^{[9]}
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 Julia broadcast multiplication, with the symbol .*
.^{[13]} In Fortran, R,^{[14]} APL, J and Wolfram Language (Mathematica), it is done through simple multiplication operator *
, 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)
.^{[15]}
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.
Applications
The Hadamard product appears in lossy compression algorithms such as JPEG. The decoding step involves an entryforentry product, in other words the Hadamard product.^{[citation needed]}
It is also used in the machine learning literature, for example to describe the architecture of recurrent neural networks as GRUs or LSTMs.^{[citation needed]}
Analogous operations
Other Hadamard operations are also seen in the mathematical literature,^{[16]} namely the Hadamard root and Hadamard power (which are in effect the same thing because of fractional indices), defined for a matrix such that:
For
and for
The Hadamard inverse reads:^{[16]}
A Hadamard division is defined as:^{[17]}^{[18]}
The penetrating face product
According to the definition of V. Slyusar the penetrating face product of the p×g matrix and ndimensional matrix (n > 1) with p×g blocks () is a matrix of size of the form:^{[19]}
Example
If
then
Main properties
 ;^{[19]}
 ,
where denotes the facesplitting product of matrices,
 , where is a vector.
Applications
The penetrating face product is used in the tensormatrix theory of digital antenna arrays.^{[19]} This operation can also be used in artificial neural network models, specifically convolutional layers.^{[20]}
See also
References
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 ^ ^{a} ^{b} Reams, Robert (1999). "Hadamard inverses, square roots and products of almost semidefinite matrices". Linear Algebra and Its Applications. 288: 35–43. doi:10.1016/S00243795(98)101623.
 ^ Wetzstein, Gordon; Lanman, Douglas; Hirsch, Matthew; Raskar, Ramesh. "Supplementary Material: Tensor Displays: Compressive Light Field Synthesis using Multilayer Displays with Directional Backlighting" (PDF). MIT Media Lab.
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 ^ ^{a} ^{b} ^{c} Slyusar, V. I. (March 13, 1998). "A Family of Face Products of Matrices and its properties" (PDF). Cybernetics and Systems Analysis C/C of Kibernetika I Sistemnyi Analiz. 1999. 35 (3): 379–384. doi:10.1007/BF02733426.
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