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Concept
Kronecker product
Summary
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 (which is denoted by the same symbol) 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 (1823–1891), 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 Johann Georg Zehfuss, who in 1858 described this matrix operation, but Kronecker product is currently the most widely used. The misattribution to Kronecker rather than Zehfuss was due to Kurt Hensel.
If A is an m × n matrix and B is a p × q matrix, then the Kronecker product A ⊗ B is the pm × qn block matrix:
more explicitly:
Using and to denote truncating integer division and remainder, respectively, and numbering the matrix elements starting from 0, one obtains and For the usual numbering starting from 1, one obtains
and
If A and B represent linear transformations V1 → W1 and V2 → W2, respectively, then the tensor product of the two maps is represented by A ⊗ B, which is the same as V1 ⊗ V2 → W1 ⊗ W2.
Similarly:
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 use the "vec trick" to rewrite this equation as
Here, vec(X) 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 AXB = C has a unique solution, if and only if A and B are invertible .
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