Kronecker product
The Kronecker product is in mathematics, a special product of two matrices of arbitrary size. The result of the Kronecker product is a large matrix obtained by considering all possible products of entries of the two output matrices. It is named after the German mathematician Leopold Kronecker.
- 4.1 Proof of equivalence
Definition
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 again a matrix, but from the dimension.
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 square, so can be selected.
The Kronecker product is associative. that is
Symmetries
For the transposition
For the complex conjugate matrix
For the adjoint matrix
References to other operations
The Kronecker product is bilinear with the matrix addition, ie
If the matrix products and defined, it shall
Parameters
Are square matrices and then for the track
For the rank
If a and a matrix then for the determinant
Are the eigenvalues of the eigenvalues of and then applies
Is then valid for the spectral norm
Inverse
Are invertible so is
Also applies to the Moore -Penrose inverse
More generally, Sind and generalized inverses of and so is a generalized inverse of.
Matrix equation
There are given the matrices
And a matrix of wanted, so that the following applies. Now, the following equivalence holds:
Here is the column-wise vectorization of a matrix into a column vector.
The columns of the matrix as a column vector of length.
Is analogous to a column vector of length.
If you have determined the vector, so it follows directly the corresponding isomorphic matrix.
Proof of the equivalence
It is
It is
System of equations with matrix coefficients
For and the matrices are given.
We are looking for the matrices that the system of equations
Solve. This task is equivalent to solving the equation system
Other applications
The Kronecker product is for example used in the generalized linear regression analysis in order to construct a covariance matrix of correlated interference. This gives about a block diagonal Zellnermatrix here.
In addition, you need the Kronecker product in quantum mechanics to systems with several particles, which have a limited range on both sides to describe. States of several particles are then Kronecker products of single-particle states. In case of an unbounded spectrum only the algebraic structure of a Kronecker product is maintained since then no representation by matrices exist.
Related to tensor products
Given two linear maps between finite dimensional vector spaces. Then there is always exactly one linear mapping
Between the tensor products with
If we choose the vector spaces and one base, so we can assign the picture its representation matrix is the matrix representation of. The Kronecker product of the representation matrices is now exactly the representation matrix of the tensorierten figure.
For this to work, but the bases have to be selected and to correct: If the selected base and the base is given by, so we take as a basis for the tensor product. Analog for.
Historical
The Kronecker product is named after Leopold Kronecker, apparently because he defined it as the first and used. Previously, the Kronecker product is sometimes called Zehfuß matrix, after Johann Georg Zehfuß.