Vech Function

Definition

Let $A$ : $n\times n$ symmetric matrix

Then vech function (half-vectorization) stacks the lower triangular elements column-by-column: $\mathrm{vech}(A)=\left[\begin{array}{c}{a}_{11}\\ a_{21}\\ ⋮\\ a_{n1}\\ a_{22}\\ a_{32}\\ ⋮\\ a_{n2}\\ a_{33}\\ ⋮\\ a_{nn}\end{array}\right]$

The vech function is a mapping of $R^{n\times n}\mapsto R^{n(n+1)/2}$

Example

$$\mathrm{vech}\left(\left[\begin{array}{ccc}1& 2& 3\\ 2& 4& 5\\ 3& 5& 6\end{array}\right]\right)=\left[\begin{array}{c}1\\ 2\\ 3\\ 4\\ 5\\ 6\end{array}\right]$$