comm.prod

Matrix multiplication envolving the commutation matrix

Description

Given the row and column dimension of a commutation and matrix x , performs one of the matrix-matrix operations:

• \(\mathbf{Y} = \mathbf{KX}\) , if side = "left" and transposed = FALSE , or
• \(\mathbf{Y} = \mathbf{K}^T\mathbf{X}\) , if side = "left" and transposed = TRUE , or
• \(\mathbf{Y} = \mathbf{XK}\) , if side = "right" and transposed = FALSE , or
• \(\mathbf{Y} = \mathbf{XK}^T\) , if side = "right" and transposed = TRUE ,

where \(\mathbf{K}\) is the commutation matrix of order \(mn\) . The main aim of comm.prod is to do this matrix multiplication without forming the commutation matrix.

Usage

comm.prod(m = 1, n = m, x = NULL, transposed = FALSE, side = "left")

Arguments

Argument	Description
m	a positive integer row dimension.
n	a positive integer column dimension.
x	numeric matrix (or vector).
transposed	logical. Commutation matrix should be transposed?
side	a string selecting if commutation matrix is pre-multiplying x , that is side = "left" or post-multiplying x , by using side = "right" .

Details

Underlying Fortran code only uses information provided by comm.info to performs the matrix multiplication. The commutation matrix is never created.

Seealso

commutation

Examples

K42 <- commutation(m = 4, n = 2, matrix = TRUE)
x <- matrix(1:24, ncol = 3)
y <- K42 %*% x

z <- comm.prod(m = 4, n = 2, x) # K42 is not stored
all(z == y) # matrices y and z are equal!