Symmetricvectorization

Symmetricvectorization is the process of converting a symmetric matrix into a vector representation that exploits the matrix’s symmetry to remove redundancy. In various contexts it appears under different names, notably vech and svec, with conventions that differ about which triangle is stacked and whether off-diagonal entries are scaled.

One common form is vech, the vectorization of the lower triangular part of a symmetric n-by-n matrix S, including the diagonal. This yields a vector in R^{n(n+1)/2}. For example, for S = [a b c; b d e; c e f], vech(S) = [a, b, c, d, e, f]^T.

Another widely used form is svec, the symmetric vectorization that stacks the entries from the upper triangle (or lower triangle) but scales off-diagonal elements by sqrt(2) to preserve the Frobenius inner product. For the same S, svec(S) = [a, sqrt(2) b, sqrt(2) c, d, sqrt(2) e, f]^T. This scaling ensures that inner products of symmetric matrices correspond to standard Euclidean inner products.

Relationship to ordinary vec: for any symmetric S, vec(S) = D vec(S) = D vech(S), where D is the duplication matrix that maps vech(S) to vec(S). Equivalently, vech(S) = L vec(S), where L is an elimination matrix. These mappings enable efficient storage and computation in algorithms that operate on symmetric matrices, such as linear least squares, eigenvalue problems, and semidefinite programming.

Applications and considerations: symmetricvectorization reduces memory usage and can simplify optimization and algebraic manipulation by avoiding duplicate entries present in standard vec(S). Different conventions (vech vs. svec) are chosen to suit specific mathematical or computational requirements. See also duplication matrix and related operators.