fastmatrix (version 0.3-8)

commutation: Commutation matrix

Description

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

Usage

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

Arguments

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?

Value

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

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.

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.

See Also

comm.info

Examples

# NOT RUN {
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!
# }