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
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 nonsingular (Horn & Johnson 1991, Lemma 4.3.1).
If X is row-ordered into the column vector x then AXB can be also be written as(Jain 1989, 2.8 Block Matrices and Kronecker Products) (A ⊗ BT)x.
Read more about this topic: Kronecker Product
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