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Encyclopedia > Kronecker product

In mathematics, the Kronecker product, denoted by otimes, is an operation on two matrices of arbitrary size resulting in a block matrix. It is a special case of a tensor product. The Kronecker product should not be confused with the usual matrix multiplication, which is an entirely different operation. In mathematics, a matrix (plural matrices) is a rectangular table of numbers or, more generally, a table consisting of abstract quantities that can be added and multiplied. ... In the mathematical discipline of matrix theory, a block matrix or a partitioned matrix is a partition of a matrix into rectangular smaller matrices called blocks. ... In mathematics, the tensor product, denoted by , may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras and modules. ... This article gives an overview of the various ways to multiply matrices. ...

Contents


Definition

If A is an m-by-n matrix and B is a p-by-q matrix, then the Kronecker product A otimes B is the mp-by-nq block matrix

A otimes B = begin{bmatrix} a_{11} B & cdots & a_{1n}B vdots & ddots & vdots a_{m1} B & cdots & a_{mn} B end{bmatrix}.

More explicitly, we have

A otimes B = begin{bmatrix} a_{11} b_{11} & a_{11} b_{12} & cdots & a_{11} b_{1q} & cdots & cdots & a_{1n} b_{11} & a_{1n} b_{12} & cdots & a_{1n} b_{1q} a_{11} b_{21} & a_{11} b_{22} & cdots & a_{11} b_{2q} & cdots & cdots & a_{1n} b_{21} & a_{1n} b_{22} & cdots & a_{1n} b_{2q} vdots & vdots & ddots & vdots & & & vdots & vdots & ddots & vdots a_{11} b_{p1} & a_{11} b_{p2} & cdots & a_{11} b_{pq} & cdots & cdots & a_{1n} b_{p1} & a_{1n} b_{p2} & cdots & a_{1n} b_{pq} vdots & vdots & & vdots & ddots & & vdots & vdots & & vdots vdots & vdots & & vdots & & ddots & vdots & vdots & & vdots a_{m1} b_{11} & a_{m1} b_{12} & cdots & a_{m1} b_{1q} & cdots & cdots & a_{mn} b_{11} & a_{mn} b_{12} & cdots & a_{mn} b_{1q} a_{m1} b_{21} & a_{m1} b_{22} & cdots & a_{m1} b_{2q} & cdots & cdots & a_{mn} b_{21} & a_{mn} b_{22} & cdots & a_{mn} b_{2q} vdots & vdots & ddots & vdots & & & vdots & vdots & ddots & vdots a_{m1} b_{p1} & a_{m1} b_{p2} & cdots & a_{m1} b_{pq} & cdots & cdots & a_{mn} b_{p1} & a_{mn} b_{p2} & cdots & a_{mn} b_{pq} end{bmatrix}.

Examples

begin{bmatrix} 1 & 2 3 & 1 end{bmatrix} otimes begin{bmatrix} 0 & 3 2 & 1 end{bmatrix} = begin{bmatrix} 1cdot 0 & 1cdot 3 & 2cdot 0 & 2cdot 3 1cdot 2 & 1cdot 1 & 2cdot 2 & 2cdot 1 3cdot 0 & 3cdot 3 & 1cdot 0 & 1cdot 3 3cdot 2 & 3cdot 1 & 1cdot 2 & 1cdot 1 end{bmatrix} = begin{bmatrix} 0 & 3 & 0 & 6 2 & 1 & 4 & 2 0 & 9 & 0 & 3 6 & 3 & 2 & 1 end{bmatrix}.
begin{bmatrix} a_{11} & a_{12} a_{21} & a_{22} a_{31} & a_{32} end{bmatrix} otimes begin{bmatrix} b_{11} & b_{12} & b_{13} b_{21} & b_{22} & b_{23} end{bmatrix} = begin{bmatrix} a_{11} b_{11} & a_{11} b_{12} & a_{11} b_{13} & a_{12} b_{11} & a_{12} b_{12} & a_{12} b_{13} a_{11} b_{21} & a_{11} b_{22} & a_{11} b_{23} & a_{12} b_{21} & a_{12} b_{22} & a_{12} b_{23} a_{21} b_{11} & a_{21} b_{12} & a_{21} b_{13} & a_{22} b_{11} & a_{22} b_{12} & a_{22} b_{13} a_{21} b_{21} & a_{21} b_{22} & a_{21} b_{23} & a_{22} b_{21} & a_{22} b_{22} & a_{22} b_{23} a_{31} b_{11} & a_{31} b_{12} & a_{31} b_{13} & a_{32} b_{11} & a_{32} b_{12} & a_{32} b_{13} a_{31} b_{21} & a_{31} b_{22} & a_{31} b_{23} & a_{32} b_{21} & a_{32} b_{22} & a_{32} b_{23} end{bmatrix}.

Properties

Bilinearity and associativity

The Kronecker product is a special case of the tensor product, so it is bilinear and associative: In mathematics, a bilinear operator is a generalized multiplication which satisfies the distributive law. ... In mathematics, associativity is a property that a binary operation can have. ...

A otimes (B+C) = A otimes B + A otimes C qquad mbox{(if } B mbox{ and } C mbox{ have the same size)},
(A+B) otimes C = A otimes C + B otimes C qquad mbox{(if } A mbox{ and } B mbox{ have the same size)},
(kA) otimes B = A otimes (kB) = k(A otimes B),
(A otimes B) otimes C = A otimes (B otimes C),

where A, B and C are matrices and k is a scalar.


The Kronecker product is not commutative: in general, A otimes B and B otimes A are different matrices. However, A otimes B and B otimes A are permutation equivalent, meaning that there exist permutation matrices P and Q such that Mathematical meaning A map or binary operation is said to be commutative when, for any x in A and any y in B . ... In linear algebra, a permutation matrix is a binary matrix that has exactly one entry 1 in each row and each column and 0s elsewhere. ...

A otimes B = P , (B otimes A) , Q.

If A and B are square matrices, then A otimes B and B otimes A are even permutation similar, meaning that we can take P = QT. Several equivalence relations in mathematics are called similarity. ...


The mixed-product property

If A, B, C and D are matrices of such size that one can form the matrix products AC and BD, then

(A otimes B)(C otimes D) = AC otimes BD.

This is called the mixed-product property, because it mixes the ordinary matrix product and the Kronecker product. It follows that A otimes B is invertible if and only if A and B are invertible, in which case the inverse is given by In mathematics and especially linear algebra, an n-by-n matrix A is called invertible, non-singular or regular if there exists another n-by-n matrix B such that AB = BA = In, where In denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication. ...

(A otimes B)^{-1} = A^{-1} otimes B^{-1}.

Spectrum

Suppose that A and B are square matrices of size n and q respectively. Let λ1, ..., λn be the eigenvalues of A and μ1, ..., μq be those of B (listed according to multiplicity). Then the eigenvalues of A otimes B are In mathematics, a number is called an eigenvalue of a matrix if there exists a nonzero vector such that the matrix times the vector is equal to the same vector multiplied by the eigenvalue. ...

lambda_i mu_j, qquad i=1,ldots,n ,, j=1,ldots,q.

It follows that the trace and determinant of a Kronecker product are given by In linear algebra, the trace of an n-by-n square matrix A is defined to be the sum of the elements on the main diagonal (the diagonal from the upper left to the lower right) of A, i. ... In algebra, a determinant is a function depending on n that associates a scalar det(A) to every n×n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ...

operatorname{tr}(A otimes B) = operatorname{tr} A , operatorname{tr} B quadmbox{and}quad det(A otimes B) = (det A)^q (det B)^n.

Singular values

If A and B are rectangular matrices, then one can consider their singular values. Suppose that A has rA nonzero singular values, namely In linear algebra singular value decomposition (SVD) is an important factorization of a rectangular real or complex matrix, with several applications in signal processing and statistics. ...

sigma_{A,i}, qquad i = 1, ldots, r_A.

Similarly, denote the nonzero singular values of B by

sigma_{B,i}, qquad i = 1, ldots, r_B.

Then the Kronecker product A otimes B has rArB nonzero singular values, namely

sigma_{A,i} sigma_{B,j}, qquad i=1,ldots,r_A ,, j=1,ldots,r_B.

Since the rank of a matrix equals the number of nonzero singular values, we find that In linear algebra, the column rank (row rank respectively) of a matrix A with entries in some field is defined to be the maximal number of columns (rows respectively) of A which are linearly independent. ...

operatorname{rank}(A otimes B) = operatorname{rank} A , operatorname{rank} B.

Relation to the abstract tensor product

The Kronecker product of matrices corresponds to the abstract tensor product of linear maps. Specifically, if the matrices A and B represent linear transformations V1 → W1 and V2 → W2, respectively, then the matrix A otimes B represents the tensor product of the two maps, V1 otimes V2 → W1 otimes W2.


Matrix equations

The Kronecker product can be used to get a convenient representation for some matrix equations. Consider for instance the equation AXB = C, where A, B and C are given matrices and the matrix X is the unknown. We can rewrite this equation as

(B^top otimes A) , operatorname{vec} X = operatorname{vec} C.

It now follows from the properties of the Kronecker product that the equation AXB = C has a unique solution if and only if A and B are nonsingular.


Here, vec X denotes the vector formed by collecting the entries of the matrix X in one long vector. Specifically, if X is an m-by-n matrix, then

operatorname{vec} X = [ x_{11}, x_{21}, ldots, x_{m1}, x_{12}, x_{22}, ldots, x_{m2}, ldots, x_{1n}, x_{2n}, ldots, x_{mn} ]^top.

History

The Kronecker product is named after Leopold Kronecker, even though there is little evidence that he was the first to define and use it. Indeed, in the past the Kronecker product was sometimes called the Zehfuss matrix, after Johann Georg Zehfuss. Leopold Kronecker Leopold Kronecker (December 7, 1823 - December 29, 1891) was a German mathematician and logician who argued that arithmetic and analysis must be founded on whole numbers, saying, God made the integers; all else is the work of man (Bell 1986, p. ...


External links

PlanetMath is a free, collaborative, online mathematics encyclopedia. ...

References

  • Roger Horn and Charles Johnson. Topics in Matrix Analysis, Chapter 4. Cambridge University Press, 1991. ISBN 0-521-46713-6.

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Kronecker Product of Matrices and Applications, Wissenschaftsverlag, 1991.
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The tensor product inherits all the indices of its factors.
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