matrix-op: vech, tr and other matrix operators

In sn: The Skew-Normal and Related Distributions Such as the Skew-t and the SUN

vech, tr and other matrix operators

Description

vech and other matrix operators

Usage

vech(A)
vech2mat(v)
duplicationMatrix(n)
tr(x)
blockDiag(...)

Arguments

A	a (symmetric) square numeric 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.
x	a square numeric matrix.
...	an abitrary numer of matrices or objects coercible into matrices.

Value

a vector in case of vech, a scalar in case of tr, otherwise a matrix.

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).

For a square numeric matrix x, tr(x) returns its trace.

blockDiag(...) creates a block-diagonal matrix from a set of matrices or objects coercible into matrices. Generally, this is useful only for numeric objects.

Author

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

References

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

Examples

M <- toeplitz(1:4)
v <- vech(M)
vech2mat(v) - M
D <- duplicationMatrix(ncol(M))
# D %*% vech(M) - as.vector(M), must be a one-column matrix of 0s
tr(outer(1:4,2:5))
blockDiag(M[1:2,], 1:2, diag(5:6))

sn documentation built on April 5, 2023, 5:15 p.m.