The half-vectorization, or vech operator, of an 
symmetric matrix 
is the 
column vector
obtained by stacking the entries on and below the main diagonal,
 |
Some authors instead stack the upper triangle, so the convention must be stated. For a symmetric matrix, the omitted entries are duplicates of retained entries and no information is lost.
Half-vectorization is a compressed form of matrix vectorization. If 
stacks every column of 
, then matrices called the duplication and elimination matrices
convert between 
and 
.
The operator is useful in matrix calculus and in multivariate statistics, particularly
for covariance matrices.
