Khatri–Rao product

is defined as[1][2][3] in which the ij-th block is the mipi × njqj 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 size of the product is then (Σi mipi) × (Σj njqj).

This product assumes the partitions of the matrices are their columns.

The resulting product is a mp × n matrix of which each column is the Kronecker product of the corresponding columns of A and B.

Using the matrices from the previous examples with the columns partitioned: so that: This column-wise version of the Khatri–Rao product is useful in linear algebra approaches to data analytical processing[5] and in optimizing the solution of inverse problems dealing with a diagonal matrix.

[6][7] In 1996 the column-wise Khatri–Rao product was proposed to estimate the angles of arrival (AOAs) and delays of multipath signals[8] and four coordinates of signals sources[9] at a digital antenna array.

An alternative concept of the matrix product, which uses row-wise splitting of matrices with a given quantity of rows, was proposed by V. Slyusar[10] in 1996.

This type of operation is based on row-by-row Kronecker products of two matrices.

Using the matrices from the previous examples with the rows partitioned: the result can be obtained:[9][11][13] where A, B and C are matrices, and k is a scalar, Source:[18] Source:[18] If

are independent components a random matrix

with independent identically distributed rows

According to the definition of V. Slyusar[9][13] the block face-splitting product of two partitioned matrices with a given quantity of rows in blocks can be written as : The transposed block face-splitting product (or Block column-wise version of the Khatri–Rao product) of two partitioned matrices with a given quantity of columns in blocks has a view:[9][13] The Face-splitting product and the Block Face-splitting product used in the tensor-matrix theory of digital antenna arrays.