Hadamard product (matrices)
In mathematics, the Hadamard product (also known as the Schur product or the entrywise product) is a binary operation that takes two matrices of the same dimensions, and produces another matrix where each element ij is the product of elements ij of the original two matrices. It should not be confused with the more common matrix product. It is attributed to, and named after, either French mathematician Jacques Hadamard, or German mathematician Issai Schur.
For two matrices, , of the same dimension, , the Hadamard product, , is a matrix, of the same dimension as the operands, with elements given by
For matrices of different dimensions ( and , where or or both) the Hadamard product is undefined.
For example, the Hadamard product for a 3×3 matrix A with a 3×3 matrix B is:
The identity matrix under Hadamard multiplication of two m-by-n matrices is an m-by-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 and , and corresponding diagonal matrices and with these vectors as their leading diagonals, the following identity holds:
where denotes the conjugate transpose of . In particular, using vectors of ones, this shows that the sum of all elements in the Hadamard product is the trace of . A related result for square and , is that the row-sums of their Hadamard product are the diagonal elements of :
The Hadamard product satisfies the rank inequality
Schur product theorem
The Hadamard product of two positive-semidefinite matrices is positive-semidefinite. This is known as the Schur product theorem, after German mathematician Issai Schur. For positive-semidefinite matrices A and B, it is also known that
In programming languages
Hadamard multiplication is built into certain programming languages under various names. In MATLAB, GNU Octave and GAUSS, it is known as array multiplication, with the symbol
.*. In Fortran, R, J and Wolfram Language (Mathematica), it is done through simple multiplication operator
*, whereas the matrix product is done through the function
+/ .* and the
. operators, respectively. In Python with the numpy numerical library or the sympy symbolic library, multiplication of
array objects as
a1*a2 produces the Hadamard product, but with otherwise
m1*m2 will produce a matrix product. The Eigen C++ library provides a
cwiseProduct member function for the
Matrix class (
a.cwiseProduct(b)), while the Armadillo library use the operator
% to make compact expressions (
a % b;
a * b is a matrix product).
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:
- Davis, Chandler. "The norm of the Schur product operation." Numerische Mathematik 4.1 (1962): 343-344.
- (Horn & Johnson 1985, Ch. 5)
- Million, Elizabeth. "The Hadamard Product" (PDF). Retrieved 2 January 2012.
- (Horn & Johnson 1991)
- (Styan 1973)
- "Arithmetic Operators + - * / \ ^ ' -". MATLAB documentation. MathWorks. Retrieved 2 January 2012.
- "Matrix multiplication". An Introduction to R. The R Project for Statistical Computing. 16 May 2013. Retrieved 24 August 2013.
- Reams, Robert (1999). "Hadamard inverses, square roots and products of almost semidefinite matrices". Linear Algebra and its Applications 288: 35–43. doi:10.1016/s0024-3795(98)10162-3.