This article deals with the Kronecker product of matrices, for the Kronecker product of cohomology and homology classes see
Kronecker pairing .
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 .
definition
If there is a matrix and a matrix, the Kronecker product is defined as
Explicit:
-
.
That is, each element of the matrix is multiplied by the matrix . So the result is a matrix with rows and columns.
example
properties
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
-
.
Symmetries
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
-
,
and
If the matrix products and are defined, then applies
-
.
Parameters
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
-
.
Inverse
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 .
Historical
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
swell
-
^ Willi Hans Steeb : Kronecker Product of Matrices and Applications . BI-Wiss. Verlag, 1991, ISBN 3-411-14811-X , p. 16
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^ Walter Strobl, "Georg Zehfuss: His life and his works", online