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
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,
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).
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)
providing 
.