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In mathematics, the Kronecker product, denoted by , 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. It is named after German mathematician Leopold Kronecker.

Definition

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

More explicitly, we have

Examples

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Properties

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Bilinearity and associativity

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The Kronecker product is a special case of the tensor product, so it is bilinear and associative:

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

The Kronecker product is not commutative: in general, A B and B A are different matrices. However, A B and B A are permutation equivalent, meaning that there exist permutation matrices P and Q such that

If A and B are square matrices, then A B and B A are even permutation similar, meaning that we can take P = QT.

The mixed-product property

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

This is called the mixed-product property, because it mixes the ordinary matrix product and the Kronecker product. It follows that A B is invertible if and only if A and B are invertible, in which case the inverse is given by

Kronecker sum and exponentiation

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If A is n-by-n, B is m-by-m and denotes the k-by-k identity matrix then we can define the Kronecker sum, , by

We have the following formula for the matrix exponential which is useful in the numerical evaluation of certain continuous-time Markov processes,

Spectrum

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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 B are

It follows that the trace and determinant of a Kronecker product are given by

Singular values

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If A and B are rectangular matrices, then one can consider their singular values. Suppose that A has rA nonzero singular values, namely

Similarly, denote the nonzero singular values of B by

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

Since the rank of a matrix equals the number of nonzero singular values, we find that

Relation to the abstract tensor product

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The Kronecker product of matrices corresponds to the abstract tensor product of linear maps. Specifically, if the matrices A and B represent linear transformations V1W1 and V2W2, respectively, then the matrix A B represents the tensor product of the two maps, V1 V2W1 W2.

Relation to products of graphs

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The Kronecker product of the adjacency matrices of two graphs is the adjacency matrix of the tensor product graph. The Kronecker sum of the adjacency matrices of two graphs is the adjacency matrix of the Cartesian product graph. See [1], answer to Exercise 96.

Transpose

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The operation of transposition is distributive over the Kronecker product:

Matrix equations

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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

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 (Horn & Johnson 1991, Lemma 4.3.1).

Here, vec(X) denotes the vectorization of the matrix X formed by stacking the columns of X into a single column vector.

If X is row-ordered into the column vector x then can be also be written as (Jain 1989, 2.8 block Matrices and Kronecker Products)

History

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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.

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Two related matrix operators are the Tracy-Singh and Khatri-Rao products which operate on partitioned matrices. Let the -by- matrix be partitioned into the -by- blocks and -by- matrix into the -by- blocks Bkl with of course , , and We then define the Tracy-Singh product to be

which means that the th subblock of the -by- product is the -by- matrix , of wich the th subblock equals the -by- matrix . For example, if and both are -by- partitioned matrices e.g.:

we get:

The Khatri-Rao product is defined as

in which the th block is the -by--sized Kronecker product of the corresponding blocks of and , assuming that the 'horizontal' and 'vertical' number of subblocks of both matrices is equal. The size of the product is then -by-. Proceeding with the same matrices as the previous example we obtain:

A column-wise Kronecker product,also called the Khatri-Rao product of two matrices assumes the partitions of the matrices as their columns. In this case , , and . The resulting product is a -by- matrix of which each column is the Kronecker product of the corresponding columns of and . We can only use the matrices from the previous examples if we change the partitions:

so that:

References

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  1. ^ D. E. Knuth: "Pre-Fascicle 0a: Introduction to Combinatorial Algorithms", zeroth printing (revision 2), to appear as part of D.E. Knuth: The Art of Computer Programming Vol. 4A
  • Horn, Roger A.; Johnson, Charles R. (1991), Topics in Matrix Analysis, Cambridge University Press, ISBN 0-521-46713-6.
  • Jain, Anil K. (1989), Fundamentals of Digital Image Processing, Prentice Hall, ISBN 0-13-336165-9.
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