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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 denotes the k-by-k identity matrix then the Kronecker sum is defined by:
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.
In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers.The function is 1 if the variables are equal, and 0 otherwise: = {, =. or with use of Iverson brackets: = [=] For example, = because , whereas = because =.
The motivation for the use of the Kronecker sum in this definition comes from the case in which and come from representations and of a Lie group. In that case, a simple computation shows that the Lie algebra representation associated to Π 1 ⊗ Π 2 {\displaystyle \Pi _{1}\otimes \Pi _{2}} is given by the preceding formula.
The matrix form of the separation of variables is the Kronecker sum. As an example we consider the 2D discrete Laplacian on a regular grid:
For example, for the 2×2 matrix = [] , the vectorization is = []. The ... Alternatively, the linear sum can be expressed using the Kronecker product: ...
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.