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Hadamard product (matrices)

Definition

For two matrices, ${\displaystyle A,B}$, of the same dimension, ${\displaystyle m\times n}$ the Hadamard product, ${\displaystyle A\circ B}$, is a matrix, of the same dimension as the operands, with elements given by

${\displaystyle (A\circ B)_{i,j}=($A)_{i,j}\cdot (B)_{i,j}} .

For matrices of different dimensions (${\displaystyle m\times n}$ and ${\displaystyle p\times q}$, where ${\displaystyle m\not =p}$or${\displaystyle n\not =q}$ or both) the Hadamard product is undefined.

Properties

The Hadamard product is commutative, associative and distributive over addition. That is,

${\displaystyle A\circ B=B\circ A,}$

${\displaystyle A\circ (B\circ C)=(A\circ B)\circ C,}$

${\displaystyle A\circ (B+C)=A\circ B+A\circ C.}$

The identity matrix under Hadamard multiplication of two m-by-n matrices is 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.^[3]

For vectors ${\displaystyle x}$ and ${\displaystyle y}$, and corresponding diagonal matrices ${\displaystyle D_{x}}$ and ${\displaystyle D_{y}}$ with these vectors as their leading diagonals, the following identity holds:^[4]

${\displaystyle x^{*}(A\circ B)y=\mathrm {tr} (D_{x}^{*}AD_{y}B^{T})}$,

where ${\displaystyle x^{*}}$ denotes the conjugate transposeof${\displaystyle x}$. In particular, using vectors of ones, this shows that the sum of all elements in the Hadamard product is the traceof${\displaystyle AB^{T}}$. A related result for square ${\displaystyle A}$ and ${\displaystyle B}$, is that the row-sums of their Hadamard product are the diagonal elements of ${\displaystyle AB^{T}:}$^[5]

${\displaystyle \sum _{j}(A\circ B)_{i,j}=(AB^{T})_{i,i}.}$

The Hadamard product is a principal submatrix of the Kronecker product.

Schur product theorem

The Hadamard product of two positive-semidefinite matrices is positive-semidefinite.^[5] This is known as the Schur product theorem,^[3] after German mathematician Issai Schur. For positive-semidefinite matrices A and B, it is also known that

${\displaystyle \det(A\circ B)\geq \det($A)\det(B).\,} ^[5]

In programming languages

Hadamard multiplication is built into certain programming languages under various names. In MATLAB, it is known as array multiplication, with the symbol .*.^[6] GAUSS uses the same approach. In Fortran and Mathematica, it is done through simple multiplication operator *, whereas the matrix product is done through the function matmul and the . operator, 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 identical matrix objects m1*m2 will produce a matrix product. (There are mappings between arrays and matrices: sympy and numpy provide a .A attribute for matrix objects, sympy provides a .M attribute for array objects, and numpy.asmatrix(a1) will produce a matrix view of the array a1.) 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). In R the Hadamard product is computed by default, with matrix.A%*%matrix.B giving the standard matrix product.^[7]

Applications

The Hadamard product appears in lossy compression algorithms such as JPEG. The decoding step involves an entry-for-entry product, i.e., Hadamard product.

See also

• Pointwise product

References