Kronecker product

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In mathematics, the Kronecker product is a special product of two matrices of any size. The result of the Kronecker product is a large matrix that is created by considering all possible products of entries in the two output matrices. It is named after the German mathematician Leopold Kronecker .


If there is a matrix and a matrix, the Kronecker product is defined as



That is, each element of the matrix is multiplied by the matrix . So the result is a matrix with rows and columns.



Calculation rules

The Kronecker product is not commutative , that is, in general


However, there are permutation matrices such that

applies. Are there and are square, you can choose .

The Kronecker product is associative . That is, it applies



The following applies to the transposition


The following applies to the conjugate matrix


The following applies to the adjoint matrix


References to other operations

The Kronecker product is bilinear with the matrix addition , that is, it applies



If the matrix products and are defined, then applies



If and are square matrices, then applies to the trace


The following applies to the rank


If one is and a matrix, then holds for the determinant


If the eigenvalues of and are the eigenvalues ​​of , then:

are the eigenvalues ​​of .

The following applies to the spectral standard



If invertible , then invertible is also possible with the inverse


The following also applies to the Moore-Penrose inverse


More generally, if and are generalized inverses of and , then is a generalized inverse of .

Matrix equation

Let the matrices be given and a matrix sought such that it holds. Then the following equivalence applies:


Here stands for the column-wise vectorization of a matrix to a column vector: If the columns of the matrix are , then is

a column vector of length . Analog is a column vector of length .

Once the vector has been determined, the associated isomorphic matrix results directly from it .

Proof of equivalence

It is .

It is .

System of equations with matrix coefficients

For and let the matrices be given.

We are looking for the matrices that form the system of equations

to solve. This task is equivalent to solving the system of equations

Other uses

The Kronecker product is used, for example, in generalized linear regression models to construct a covariance matrix of correlated disturbance variables (e.g. the covariance matrix for seemingly unconnected regression equations, see covariance matrix # covariance matrix for apparently unrelated regression equations ). A block diagonal cellular matrix is obtained here .

In addition, the Kronecker product is needed in quantum mechanics to describe systems with several particles that have a spectrum that is limited on both sides. States of several particles are then Kronecker products of the single-particle states. In the case of an unrestricted spectrum, only the algebraic structure of a Kronecker product remains, since then there is no representation by matrices.

Connection with tensor products

Given are two linear mappings and between finite-dimensional vector spaces . Then there is always exactly one linear mapping

between the tensor products with


If we on the vector spaces and select one each base so we can figure their representation matrix to assign. Let it be the representation matrix of .

The Kronecker product of the representation matrices now corresponds exactly to the representation matrix of the tensored mapping , if one takes the basis on and , which results from the lexicographically arranged pairs of basis vectors of the vector spaces involved in the tensor product: Are the selected basis of and the basis of , so we take

as the basis for the tensor product . Analog for .


The Kronecker product is named after Leopold Kronecker , although Georg Zehfuss defined the product as early as 1858, which is why the Kronecker product is sometimes also called the Zehfuss product.

Web links


  1. ^ Willi Hans Steeb : Kronecker Product of Matrices and Applications . BI-Wiss. Verlag, 1991, ISBN 3-411-14811-X , p. 16
  2. ^ Walter Strobl, "Georg Zehfuss: His life and his works", online