In mathematics, the Khatri–Rao product of matrices defined as in which the ij-th block is the 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 . For example, if A and B both are partitioned matrices e.g.: we obtain: This is a submatrix of the Tracy–Singh product of the two matrices (each partition in this example is a partition in a corner of the Tracy–Singh product) and also may be called the block Kronecker product. Column-wise Kronecker product A column-wise Kronecker product of two matrices may also be called the Khatri–Rao product. This product assumes the partitions of the matrices are their columns. In this case , , and for each j: . The resulting product is a 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 and in optimizing the solution of inverse problems dealing with a diagonal matrix. In 1996 the Column-wise Khatri–Rao product was proposed to estimate the angles of arrival (AOAs) and delays of multipath signals and four coordinates of signals sources at a digital antenna array. Face-splitting product The alternative concept of the matrix product, which uses row-wise splitting of matrices with a given quantity of rows, was proposed by V. Slyusar in 1996. This matrix operation was named the "face-splitting product" of matrices or the "transposed Khatri–Rao product". 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: Main properties Examples Theorem If , where are independent components a random matrix with independent identically distributed rows , such that and , then for any vector with probability if the quantity of rows In particular, if the entries of are can get which matches the Johnson–Lindenstrauss lemma of when is small. Block face-splitting product According to the definition of V. Slyusar 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