Khatri–Rao product


In mathematics, the Khatri–Rao product or block Kronecker product of two partitioned matrices and is 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.

Column-wise Kronecker product

The column-wise Kronecker product of two matrices is a special case of the Khatri-Rao product as defined above, and 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 and delays of multipath signals and four coordinates of signals sources at a digital antenna array.

Face-splitting product

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 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

In the following properties, the operator denotes the row-wise Kronecker product and the operator denotes the column-wise Kronecker product

Examples

Source:

Theorem

Source:
If, where are independent components a random matrix with independent identically distributed rows, such that
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 of two partitioned matrices with a given quantity of columns in blocks has a view:

Main properties

  1. Transpose:
  2. :

    Applications

The Face-splitting product and the Block Face-splitting product used in the tensor-matrix theory of digital antenna arrays. These operations are also used in: