Hadamard product (matrices)

Definition

For two matrices A and B of the same dimension m × n, the Hadamard product A \odot B (sometimes A \circ B) is a matrix of the same dimension as the operands, with elements given by :(A \odot B){ij} = (A){ij} (B)_{ij}.

For matrices of different dimensions (m × n and p × q, where m ≠ p or n ≠ q), the Hadamard product is undefined.

ExampleAn example of the Hadamard product for two arbitrary 2 × 3 matrices: : \begin{bmatrix} 2 & 3 & 1 \ 0 & 8 & -2 \end{bmatrix} \odot \begin{bmatrix} 3 & 1 & 4 \ 7 & 9 & 5 \end{bmatrix} = \begin{bmatrix} 2 \times 3 & 3 \times 1 & 1 \times 4 \ 0 \times 7 & 8 \times 9 & -2 \times 5 \end{bmatrix} = \begin{bmatrix} 6 & 3 & 4 \ 0 & 72 & -10 \end{bmatrix}.

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: \begin{align} A \odot B &= B \odot A, \ A \odot (B \odot C) &= (A \odot B) \odot C, \ A \odot (B + C) &= A \odot B + A \odot C, \ (kA) \odot B &= A \odot (kB) = k(A \odot B), \ A \odot 0 &= 0 \odot A = 0. \end{align}
• 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 all of the elements are invertible, or equivalently over a field, 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: \mathbf{x}^* (A \odot B)\mathbf{y} = \operatorname{tr}\left({D}\mathbf{x}^* A {D}\mathbf{y} {B}^\mathsf{T}\right), 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, \operatorname{tr}\left(AB^{\mathsf T}\right) = \mathbf{1}^\mathsf{T}\left(A\odot B\right)\mathbf{1}. A related result for square A and B, is that the row-sums of their Hadamard product are the diagonal elements of ABT: \sum_i (A \odot B){ij} = \left(B^\mathsf{T} A\right){jj} = \left(AB^\mathsf{T}\right){ii}. Similarly, (\mathbf{y}\mathbf{x}^*) \odot A = D\mathbf{y} A D_\mathbf{x}^*. Furthermore, a Hadamard matrix–vector product can be expressed as (A \odot B) \mathbf{y} = \operatorname{diag}(A D_\mathbf{y} B^\mathsf{T}), where \operatorname{diag}(M) is the vector formed from the diagonals of matrix M. Taking \mathbf{y} = \mathbf{1}, this implies that (A \odot B) \mathbf{1} = \operatorname{diag}(A B^\mathsf{T})
• The Hadamard product is a principal submatrix of the Kronecker product.{{Cite journal |last1=Liu |first1=Shuangzhe |last2=Trenkler |first2=Götz |last3=Kollo |first3=Tõnu | last4=von Rosen |first4=Dietrich |last5=Baksalary |first5=Oskar Maria |date= 2023 |title=Professor Heinz Neudecker and matrix differential calculus | journal=Statistical Papers |volume=65 |issue=4 |pages=2605–2639 |language=en |doi=10.1007/s00362-023-01499-w}}
• The Hadamard product satisfies the rank inequality \operatorname{rank}(A \odot B) \leq \operatorname{rank}(A) \operatorname{rank}(B).
• If A and B are positive-definite matrices, then the following inequality involving the Hadamard product holds: \prod_{i=k}^n \lambda_i(A \odot B) \ge \prod_{i=k}^n \lambda_i(A B),\quad k = 1, \ldots, n, where λi(A) is the ith largest eigenvalue of A.
• If D and E are diagonal matrices, then \begin{align} D (A \odot B) E &= (D A E) \odot B = (D A) \odot (B E) \ &= (AE) \odot (D B) = A \odot (D B E). \end{align}
• The Hadamard product of two vectors \mathbf a and \mathbf b is the same as matrix multiplication of the corresponding diagonal matrix of one vector by the other vector: \mathbf a \odot \mathbf b = D_\mathbf{a} \mathbf b = D_\mathbf{b} \mathbf a.
• The \operatorname{diag} operator transforming a vector to a diagonal matrix may be expressed using the Hadamard product as \operatorname{diag}(\mathbf{a}) = (\mathbf{a} \mathbf{1}^T) \odot I, where \mathbf{1} is a constant vector with elements 1, and I is the identity matrix.

The mixed-product property

The Hadamard product obeys certain relationships with other matrix product operators.

• If \otimes is the Kronecker product, assuming A has the same dimensions as C and B as D, then (A \otimes B) \odot (C \otimes D) = (A \odot C) \otimes (B \odot D) .

• If \bull is the face-splitting product, then (A \bull B) \odot (C \bull D) = (A \odot C) \bull (B \odot D).

• If \ast is the column-wise Khatri–Rao product, then (A \bull B)(C \ast D) = (A C) \odot (B D).

Schur product theorem

Main article: Schur product theorem

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} \odot {B}) \ge \det({A}) \det({B}).

Analogous operations

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:

RootFor \begin{align} {B} &= {A}^{\circ 2} \ B_{ij} &= {A_{ij}}^2 \end{align}

and for \begin{align} {B} &= {A}^{\circ \frac12} \ B_{ij} &= {A_{ij}}^\frac12 \end{align}

InverseThe Hadamard inverse reads: \begin{align} {B} &= {A}^{\circ -1} \ B_{ij} &= {A_{ij}}^{-1} \end{align}

DivisionA Hadamard division is defined as:

\begin{align} {C} &= {A} \oslash {B} \ C_{ij} &= \frac{A_{ij}}{B_{ij}} \end{align}

In programming languages

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 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 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.

In the Maple programming language, the multiplication operator * is not defined for Matrices and Vectors. The matrix product is written using ., while the Hadamard product may be obtained using *~. (The ~ is a 'broadcast operator' similar to . in Julia.)

The R package matrixcalc introduces the 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 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.

The penetrating face product

According to the definition of V. Slyusar the penetrating face product of the p×g matrix {A} and n-dimensional matrix {B} (n 1) with p×g blocks ({B} = [B_n] ) is a matrix of size {B} of the form: {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].

Example

If {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} [\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].

Main properties

:{A} [\circ] {B} = {B} [\circ] {A}; :{M} \bull {M} = {M} [\circ] \left( {M} \otimes \mathbf {1}^\textsf{T}\right),

where \bull denotes the face-splitting product of matrices, :\mathbf{c} \bull {M} = \mathbf {c} [\circ] {M}, where \mathbf {c} is a vector.

Applications

The penetrating face product is used in the tensor-matrix theory of digital antenna arrays.

References

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