Kronecker product

In mathematics, the Kronecker product, denoted by ⊗, is an operation on two matrices of arbitrary size resulting in a block matrix. It is a generalization of the outer product (which is denoted by the same symbol) from vectors to matrices, and gives the matrix of the tensor product with respect to a standard choice of basis. The Kronecker product should not be confused with the usual matrix multiplication, which is an entirely different operation.The Kronecker product is a special case of the tensor product, so it is bilinear and associative: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 thatIf A, B, C and D are matrices of such size that one can form the matrix products AC and BD, thenIf A and C are matrices of the same size, B and D are matrices of the same size, thenIt follows that A ⊗ B is invertible if and only if both A and B are invertible, in which case the inverse is given byTransposition and conjugate transposition are distributive over the Kronecker product:Let A be an n × n matrix and let B be an m × m matrix. ThenIf A is n × n, B is m × m and Ik denotes the k × k identity matrix then we can define what is sometimes called the Kronecker sum, ⊕, bySuppose that A and B are square matrices of size n and m respectively. Let λ1, ..., λn be the eigenvalues of A and μ1, ..., μm be those of B (listed according to multiplicity). Then the eigenvalues of A ⊗ B areIf A and B are rectangular matrices, then one can consider their singular values. Suppose that A has rA nonzero singular values, namelyThe Kronecker product of matrices corresponds to the abstract tensor product of linear maps. Specifically, if the vector spaces V, W, X, and Y have bases {v1, ..., vm}, {w1, ..., wn}, {x1, ..., xd}, and {y1, ..., ye}, respectively, and if the matrices A and B represent the linear transformations S : V → X and T : W → Y, respectively in the appropriate bases, then the matrix A ⊗ B represents the tensor product of the two maps, S ⊗ T : V ⊗ W → X ⊗ Y with respect to the basis {v1 ⊗ w1, v1 ⊗ w2, ..., v2 ⊗ w1, ..., vm ⊗ wn} of V ⊗ W and the similarly defined basis of X ⊗ Y with the property that A ⊗ B(vi ⊗ wj) = (Avi) ⊗ (Bwj), where i and j are integers in the proper range.where ∘ {displaystyle circ } denotes the Hadamard product. In mathematics, the Kronecker product, denoted by ⊗, is an operation on two matrices of arbitrary size resulting in a block matrix. It is a generalization of the outer product (which is denoted by the same symbol) from vectors to matrices, and gives the matrix of the tensor product with respect to a standard choice of basis. The Kronecker product should not be confused with the usual matrix multiplication, which is an entirely different operation. The Kronecker product is named after 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, after Johann Georg Zehfuss who in 1858 described this matrix operation, but Kronecker product is currently the most trendy term. If A is an m × n matrix and B is a p × q matrix, then the Kronecker product A ⊗ B is the mp × nq block matrix: