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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.
The Kronecker sum is different from the direct sum, but is also denoted by ⊕. It is defined using the Kronecker product ⊗ and normal matrix addition. If A is n -by- n , B is m -by- m and I k {\displaystyle \mathbf {I} _{k}} denotes the k -by- k identity matrix then the Kronecker sum is defined by:
This is called the Kronecker sum, defined in Matrix addition#Kronecker sum and Kronecker product#Properties. The motivation for the use of the Kronecker sum in this definition comes from the case in which π 1 {\displaystyle \pi _{1}} and π 2 {\displaystyle \pi _{2}} come from representations Π 1 {\displaystyle \Pi _{1}} and Π 2 ...
In mathematics, the Kronecker sum of discrete Laplacians, named after Leopold Kronecker, is a discrete version of the separation of variables for the continuous Laplacian in a rectangular cuboid [broken anchor] domain.
The generalized Kronecker delta or multi-index Kronecker delta of order is a type (,) tensor that is completely antisymmetric in its upper indices, and also in its lower indices. Two definitions that differ by a factor of p ! {\displaystyle p!} are in use.
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.
The vectorization is frequently used together with the Kronecker product to express matrix multiplication as a linear transformation on matrices. In particular, vec ( A B C ) = ( C T ⊗ A ) vec ( B ) {\displaystyle \operatorname {vec} (ABC)=(C^{\mathrm {T} }\otimes A)\operatorname {vec} (B)} for matrices A , B , and C of dimensions k ...
In linear algebra, the trace of a square matrix A, denoted tr(A), [1] is the sum of the elements on its main diagonal, + + +.It is only defined for a square matrix (n × n).The trace of a matrix is the sum of its eigenvalues (counted with multiplicities).