`commutation`

Commutation matrix

Description

This function returns the commutation matrix of order \(mn\) which transforms, for an \(m\times n\) matrix \(\mathbf{A}\) , vec \((\mathbf{A})\) to vec \((\mathbf{A}^T)\) .

Usage

commutation(m = 1, n = m, matrix = FALSE, condensed = FALSE)

Arguments

Argument	Description
`m`	a positive integer row dimension.
`n`	a positive integer column dimension.
`matrix`	a logical indicating whether the commutation matrix will be returned.
`condensed`	logical. Information should be returned in compact form?

Details

This function is a wrapper function for the function comm.info . This function provides the minimum information required to create the commutation matrix. If option matrix = FALSE the commutation matrix is stored in two vectors containing the coordinate list of indexes for rows and columns. Option condensed = TRUE only returns vector of indexes for the rows of commutation matrix.

Warning: matrix = TRUE is not recommended, unless the order m and n be small. This matrix can require a huge amount of storage.

Value

Returns an \(mn\) by \(mn\) matrix (if requested).

References

Magnus, J.R., Neudecker, H. (1979). The commutation matrix: some properties and applications. The Annals of Statistics 7 , 381-394.

Magnus, J.R., Neudecker, H. (2007). Matrix Differential Calculus with Applications in Statistics and Econometrics , 3rd Edition. Wiley, New York.

Examples

z <- commutation(m = 100, condensed = TRUE)
object.size(z) # 40.6 Kb of storage

z <- commutation(m = 100, condensed = FALSE)
object.size(z) # 80.7 Kb of storage

K100 <- commutation(m = 100, matrix = TRUE) # time: < 2 secs
object.size(K100) # 400 Mb of storage, do not request this matrix!

# a small example
K32 <- commutation(m = 3, n = 2, matrix = TRUE)
a <- matrix(1:6, ncol = 2)
v <- K32 %*% vec(a)
all(vec(t(a)) == as.vector(v)) # vectors are equal!

commutation