Hadamard product (matrices)

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 mathematician Jacques Hadamard or German 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 A ⊙ B {\displaystyle A\odot B} (sometimes A ∘ B {\displaystyle A\circ B} ) 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.
An example of the Hadamard product for two arbitrary 2 × 3 matrices:
The Hadamard product obeys certain relationships with other matrix product operators.
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: det ( A ⊙ B ) ≥ det ( A ) det ( B ) . {\displaystyle \det({A}\odot {B})\geq \det({A})\det({B}).}
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:
For B = A ∘ 2 B i j = A i j 2 {\displaystyle {\begin{aligned}{B}&={A}^{\circ 2}\\B_{ij}&={A_{ij}}^{2}\end{aligned}}}
and for B = A ∘ 1 2 B i j = A i j 1 2 {\displaystyle {\begin{aligned}{B}&={A}^{\circ {\frac {1}{2}}}\\B_{ij}&={A_{ij}}^{\frac {1}{2}}\end{aligned}}}
The Hadamard inverse reads: B = A ∘ − 1 B i j = A i j − 1 {\displaystyle {\begin{aligned}{B}&={A}^{\circ -1}\\B_{ij}&={A_{ij}}^{-1}\end{aligned}}}
A Hadamard division is defined as:
C = A ⊘ B C i j = A i j B i j {\displaystyle {\begin{aligned}{C}&={A}\oslash {B}\\C_{ij}&={\frac {A_{ij}}{B_{ij}}}\end{aligned}}}
Most scientific or numerical programming languages include the Hadamard product, under various names.
In MATLAB, the Hadamard product is expressed as "dot multiply": a .* b, or the function call: times(a, b). It also has analogous dot operators which include, for example, the operators a .^ b and a ./ b. Because of this mechanism, it is possible to reserve * and ^ for matrix multiplication and matrix exponentials, respectively.
The programming language Julia has similar syntax as MATLAB, where Hadamard multiplication is called broadcast multiplication and also denoted with a .* b, and other operators are analogously defined element-wise, for example Hadamard powers use a .^ b. But unlike MATLAB, in Julia this "dot" syntax is generalized with a generic broadcasting operator . which can apply any function element-wise. This includes both binary operators (such as the aforementioned multiplication and exponentiation, as well as any other binary operator such as the Kronecker product), and also unary operators such as ! and √. Thus, any function in prefix notation f can be applied as f.(x).
Python does not have built-in array support, leading to inconsistent/conflicting notations. The NumPy numerical library interprets a*b or a.multiply(b) as the Hadamard product, and uses a@b or a.matmul(b) for the matrix product. With the SymPy symbolic library, multiplication of array objects as either a*b or a@b will produce the matrix product. The Hadamard product can be obtained with the method call a.multiply_elementwise(b). Some Python packages include support for Hadamard powers using methods like np.power(a, b), or the Pandas method a.pow(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).
In GAUSS, and HP Prime, the operation is known as array multiplication.
In Fortran, R, APL, J and Wolfram Language (Mathematica), the multiplication operator * or × apply the Hadamard product, whereas the matrix product is written using matmul, %*%, +.×, +/ .* and ., respectively. The R package matrixcalc introduces the 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 used in the machine learning literature, for example, to describe the architecture of recurrent neural networks as GRUs or LSTMs.
It is also used to study the statistical properties of random vectors and matrices.
According to the definition of V. Slyusar the penetrating face product of the p×g matrix A {\displaystyle {A}} and n-dimensional matrix B {\displaystyle {B}} (n > 1) with p×g blocks ( B = [ B n ] {\displaystyle {B}=[B_{n}]} ) is a matrix of size B {\displaystyle {B}} of the form: A [ ∘ ] B = [ A ∘ B 1 A ∘ B 2 ⋯ A ∘ B n ] . {\displaystyle {A}[\circ ]{B}=\left[{\begin{array}{c | c | c | c }{A}\circ {B}_{1}&{A}\circ {B}_{2}&\cdots &{A}\circ {B}_{n}\end{array}}\right].}
If A = [ 1 2 3 4 5 6 7 8 9 ] , B = [ B 1 B 2 B 3 ] = [ 1 4 7 2 8 14 3 12 21 8 20 5 10 25 40 12 30 6 2 8 3 2 4 2 7 3 9 ] {\displaystyle {A}={\begin{bmatrix}1&2&3\\4&5&6\\7&8&9\end{bmatrix}},\quad {B}=\left[{\begin{array}{c | c | c }{B}_{1}&{B}_{2}&{B}_{3}\end{array}}\right]=\left[{\begin{array}{c c c | c c c | c c c }1&4&7&2&8&14&3&12&21\\8&20&5&10&25&40&12&30&6\\2&8&3&2&4&2&7&3&9\end{array}}\right]}
then
A [ ∘ ] B = [ 1 8 21 2 16 42 3 24 63 32 100 30 40 125 240 48 150 36 14 64 27 14 32 18 49 24 81 ] . {\displaystyle {A}[\circ ]{B}=\left[{\begin{array}{c c c | c c c | c c c }1&8&21&2&16&42&3&24&63\\32&100&30&40&125&240&48&150&36\\14&64&27&14&32&18&49&24&81\end{array}}\right].}
where ∙ {\displaystyle \bullet } denotes the face-splitting product of matrices,
The penetrating face product is used in the tensor-matrix theory of digital antenna arrays. This operation can also be used in artificial neural network models, specifically convolutional layers.