| matrix-op {sn} | R Documentation |
vech and other matrix operators
vech(A) vech2mat(v) duplicationMatrix(n)
A |
a (symmetric) square matrix. |
v |
a numeric vector such that |
n |
a positive integer number; default is |
a vector in case of vech, otherwise a matrix.
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,
vect2mat(v) performs the inverse operation and
returns 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).
Adelchi Azzalini;
the original Octave code of duplicationMatrix is by Kurt Hornik.
Magnus, Jan R. and Neudecker, Heinz (1988). Matrix differential calculus with application in statistics and econometrics. Wiley series in probability and statistics.
M <- toeplitz(1:4) v <- vech(M) vech2mat(v) - M D <- duplicationMatrix(ncol(M)) # D %*% vech(M) - as.vector(M)