Hadamard product (matrices)
The Hadamard product, Schur product or component-wise product is in mathematics, a special product of two matrices of the same size. The resulting matrix results here by multiplying the corresponding entries in each of the two output matrices. The Hadamard product is associative, distributive and commutative with the matrix addition, if the underlying ring is also commutative.
The Hadamard product has some interesting properties. For example, the Hadamard product of two positive semidefinite matrices again semidefinite positive. Next, various characteristics ( such as standard, rank or spectral radius ) of the Hadamard product over the product of the respective parameters of the output matrices estimated. In comparison to the complex matrix product the Hadamard product, however, is less important in practice. It is named after the mathematicians Jacques Hadamard and Isay Schur.
- 4.1 spectral norm
- 4.2 Kronecker product
- 4.3 Induced sesquilinear
Definition
Is a ring and as well as two matrices over, then the Hadamard product is from and by
Defined. The result is thus a matrix of the same size, with each entry calculated by component-wise multiplication of the entries of the matrix with the entries of the matrix. As an operator symbol and the sign is sometimes used for the Hadamard product.
Example
The Hadamard product of two real (2 × 2) - matrices
Is given by
Properties
Computational rules
The Hadamard product inherits essentially the properties of the underlying ring. It is always associative, ie for matrices applies
And it is compatible with the multiplication of scalars, ie
If the underlying ring is commutative, so is the Hadamard product is commutative, that is
Wherein it differs from the matrix product is normally used. With the component-wise matrix addition and the distributive laws apply
Also applies to the transposed matrix of a Hadamard product
The Hadamard product of two symmetric matrices is therefore symmetrical again.
Algebraic Structures
The set of matrices over a ring forms with the matrix addition and the Hadamard product is back in a ring. Is a unitary ring with identity, then the matrix ring also has a unit element, the fuel matrix in which all elements are equal. With the fuel matrix is then valid for all matrices
If a body is, then that means a Hadamard matrix is invertible if all entries of equal to the zero element. The amount of the Hadamard invertible matrices then forms a group, where the entries of the Hadamard inverse of by
Are given. In addition, only matrices are considered over the field of real or complex numbers.
Positive definiteness
Are the square matrices positive semidefinite, so also their Hadamard product is positive semi-definite and is valid for the eigenvalues of
If is positive definite and positive semi-definite with positive diagonal entries, then the Hadamard product is positive definite. These statements go back to Isay Schur, who in 1911 first formulated.
Assessments
Spectral norm
If the square matrix is positive definite, then, for the spectral norm of a Hadamard product
Is the product of two matrices, then applies
The maximum Euclidean norm of the column vectors of is. Overall, one obtains the estimate:
These three assessments also go back to Isay Schur.
Kronecker product
The Kronecker product yields as a result a large matrix obtained by considering all possible products of entries of the two output matrices. Are the matrices, then the entries of the Hadamard product find exactly at the intersections of the columns with the rows of the corresponding Kronecker - product. The Hadamard product is thus a sub- matrix of the Kronecker product. Therefore applies to the spectral norm of a Hadamard product
And for the rank of Hadamard product
If two matrices, and only non-negative entries, then this also applies to and. Are there and square, then for the spectral radius ( the magnitude of the largest magnitude eigenvalue ) of a Hadamard product
Induced sesquilinear
For diagonal matrices (and only for this ) the Hadamard product and the usual matrix product match:
Are now desired, two (column) vectors and two diagonal matrices with entries from and on the diagonal, the following applies
Accordingly, the sesquilinear form generated by the Hadamard product, be written as a trace. From this example follows the Submultiplikativität the Frobenius norm with respect to the Hadamard product:
Programming
The Hadamard product is integrated into programming systems in different ways:
- In the numerical software package MATLAB, the Hadamard product is represented by the symbol combination. * While * for the matrix product is.
- In the Fortran programming language, the Hadamard product is realized by the simple multiplication operator *, while a separate routine stands for the matrix multiplication format msgid available.
- In the statistical software R, the Hadamard product is represented by *, while the matrix multiplication is realized by % *%.