sn (version 1.6-2)

matrix-op: vech and other matrix operators

Description

vech and other matrix operators

Usage

vech(A)
vech2mat(v)
duplicationMatrix(n)

Arguments

A

a (symmetric) square matrix.

v

a numeric vector such that length(v)=n*(n+1)/2 for some positive integer n.

n

a positive integer number; default is n=1.

Value

a vector in case of vech, otherwise a matrix.

Author

Adelchi Azzalini; the original Octave code of duplicationMatrix is by Kurt Hornik.

Details

For a square matrix A, vech(A) returns the vector formed by the lower triangular portion of the matrix, including the diagonal; usually, this only makes sense for a symmetric matrix of numeric values. If v=vech(M) where M is a symmetric numeric matrix, vech2mat(v) performs the inverse operation and returns the original matrix M; this explain the requirement on length(v). For a positive integer n, D=duplicationMatrix(n) is a matrix of dimension (n^2, n*(n+1)/2) such that D %*% vech(M) returns the vec-form of a symmetric matrix M of order n, that is, the vector which stacks the columns of M; for more information, see Section 3.8 of Magnus and Neudecker (1988).

References

Magnus, Jan R. and Neudecker, Heinz (1988). Matrix differential calculus with application in statistics and econometrics. Wiley series in probability and statistics.

Examples

# NOT RUN {
M <- toeplitz(1:4)
v <- vech(M)
vech2mat(v) - M
D <- duplicationMatrix(ncol(M))
# D %*% vech(M) - as.vector(M)
# }