fastmatrix (version 0.5-7721)

duplication: Duplication matrix

Description

This function returns the duplication matrix of order \(n\) which transforms, for a symmetric matrix \(\bold{A}\), vech\((\bold{A})\) into vec\((\bold{A})\).

Usage

duplication(n = 1, matrix = FALSE, condensed = FALSE)

Value

Returns an \(n^2\) by \(n(n + 1)/2\) matrix (if requested).

Arguments

n

order of the duplication matrix.

matrix

a logical indicating whether the duplication matrix will be returned.

condensed

logical. Information should be returned in compact form?.

Details

This function is a wrapper function for the function dupl.info. This function provides the minimum information required to create the duplication matrix. If option matrix = FALSE the duplication 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 columns of duplication matrix.

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

References

Magnus, J.R., Neudecker, H. (1980). The elimination matrix, some lemmas and applications. SIAM Journal on Algebraic Discrete Methods 1, 422-449.

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

See Also

dupl.info

Examples

z <- duplication(n = 100, condensed = TRUE)
object.size(z) # 40.5 Kb of storage

z <- duplication(n = 100, condensed = FALSE)
object.size(z) # 80.6 Kb of storage

D100 <- duplication(n = 100, matrix = TRUE)
object.size(D100) # 202 Mb of storage, do not request this matrix!

# a small example
D3 <- duplication(n = 3, matrix = TRUE)
a <- matrix(c( 1, 2, 3,
               2, 3, 4,
               3, 4, 5), nrow = 3)
upper <- vech(a)
v <- D3 %*% upper
all(vec(a) == as.vector(v)) # vectors are equal!