Vectorization (mathematics) - Half-vectorization

Half-vectorization

For a symmetric matrix A, the vector vec(A) contains more information than is strictly necessary, since the matrix is completely determined by the symmetry together with the lower triangular portion, that is, the n(n+1)/2 entries on and below the main diagonal. For such matrices, the half-vectorization is sometimes more useful than the vectorization. The half-vectorization, vech(A), of a symmetric n×n matrix A is the n(n+1)/2 × 1 column vector obtained by vectorizing only the lower triangular part of A:

vech(A) = T.

For example, for the 2×2 matrix A =, the half-vectorization is vech(A) = .

There exist unique matrices transforming the half-vectorization of a matrix to its vectorization and vice-versa called, respectively, the duplication matrix and the elimination matrix.