Kronecker product


In mathematics, the Kronecker product, sometimes denoted by ⊗, is an operation on two matrices of arbitrary size resulting in a block matrix. It is a specialization of the tensor product from vectors to matrices and gives the matrix of the tensor product linear map with respect to a standard choice of basis. The Kronecker product is to be distinguished from the usual matrix multiplication, which is an entirely different operation. The Kronecker product is also sometimes called matrix direct product.
The Kronecker product is named after the German mathematician Leopold Kronecker, even though there is little evidence that he was the first to define and use it. The Kronecker product has also been called the Zehfuss matrix, and the Zehfuss product, after, who in 1858 described this matrix operation, but Kronecker product is currently the most widely used term. The misattribution to Kronecker rather than Zehfuss was due to Kurt Hensel.

Definition

If A is an matrix and B is a matrix, then the Kronecker product is the block matrix:
more explicitly:
Using and to denote truncating integer division and remainder, respectively, and numbering the matrix elements starting from 0, one obtains
For the usual numbering starting from 1, one obtains
If A and B represent linear transformations and, respectively, then the tensor product of the two maps is a map represented by.

Examples

Similarly:

Properties

Relations to other matrix operations

Abstract properties

Matrix equations

The Kronecker product can be used to get a convenient representation for some matrix equations. Consider for instance the equation, where A, B and C are given matrices and the matrix X is the unknown. We can use the "vec trick" to rewrite this equation as
Here, vec denotes the vectorization of the matrix X, formed by stacking the columns of X into a single column vector.
It now follows from the properties of the Kronecker product that the equation has a unique solution, if and only if A and B are invertible.
If X and C are row-ordered into the column vectors u and v, respectively, then
The reason is that

Applications

For an example of the application of this formula, see the article on the Lyapunov equation. This formula also comes in handy in showing that the matrix normal distribution is a special case of the multivariate normal distribution. This formula is also useful for representing 2D image processing operations in matrix-vector form.
Another example is when a matrix can be factored as a Kronecker product, then matrix multiplication can be performed faster by using the above formula. This can be applied recursively, as done in the radix-2 FFT and the Fast Walsh–Hadamard transform. Splitting a known matrix into the Kronecker product of two smaller matrices is known as the "nearest Kronecker product" problem, and can be solved exactly by using the SVD. To split a matrix into the Kronecker product of more than two matrices, in an optimal fashion, is a difficult problem and the subject of ongoing research; some authors cast it as a tensor decomposition problem.
In conjunction with the least squares method, the Kronecker product can be used as an accurate solution to the hand–eye calibration problem.

Related matrix operations

Two related matrix operations are the Tracy–Singh and Khatri–Rao products, which operate on partitioned matrices. Let the matrix A be partitioned into the blocks Aij and matrix B into the blocks Bkl, with of course,, and.

Tracy–Singh product

The Tracy–Singh product is defined as
which means that the -th subblock of the product is the matrix, of which the -th subblock equals the matrix. Essentially the Tracy–Singh product is the pairwise Kronecker product for each pair of partitions in the two matrices.
For example, if A and B both are partitioned matrices e.g.:
we get:

Khatri–Rao product

Mixed-products properties
where denotes the Face-splitting product.
Similarly:
where and are vectors,
where and are vectors, and denotes the Hadamard product.
Similarly:
where is vector convolution and is the Fourier transform matrix,
where denotes the column-wise Khatri–Rao product.
Similarly:
where and are vectors.