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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.
In mathematics, the Khatri–Rao product or block Kronecker product of two partitioned matrices and is defined as [1] [2] [3] = in which the ij-th block is the m i p i × n j q j sized Kronecker product of the corresponding blocks of A and B, assuming the number of row and column partitions of both matrices is equal.
and the corresponding operation of symmetric functions is the usual product. Also note that the Littlewood–Richardson coefficients are the analogue of the Kronecker coefficients for representations of GL n, i.e. if we write W λ for the irreducible representation corresponding to λ (where λ has at most n parts), one gets that
The adjacency matrix of G × H is the Kronecker (tensor) product of the adjacency matrices of G and H. If a graph can be represented as a tensor product, then there may be multiple different representations (tensor products do not satisfy unique factorization) but each representation has the same number of irreducible factors.
The vectorization is frequently used together with the Kronecker product to express matrix multiplication as a linear transformation on matrices. In particular, = for matrices A, B, and C of dimensions k×l, l×m, and m×n.
A mixed state is described by a density matrix ρ, that is a non-negative trace-class operator of trace 1 on the tensor product . The partial trace of ρ with respect to the system B , denoted by ρ A {\displaystyle \rho ^{A}} , is called the reduced state of ρ on system A .
The Kronecker product implies the following factorization: C ( u ⊗ v ) = N D ( N ⊗ N ) ( u ⊗ v ) . {\displaystyle C(u\otimes v)=ND(N\otimes N)(u\otimes v).} Then it can be proved that in the two-dimensional vector logic the De Morgan's law is a law involving operators, and not only a law concerning operations: [ 6 ]
The main use of the commutation matrix, and the source of its name, is to commute the Kronecker product: for every m × n matrix A and every r × q matrix B, K ( r , m ) ( A ⊗ B ) K ( n , q ) = B ⊗ A . {\displaystyle \mathbf {K} ^{(r,m)}(\mathbf {A} \otimes \mathbf {B} )\mathbf {K} ^{(n,q)}=\mathbf {B} \otimes \mathbf {A} .}