svec

Svec, short for symmetric vectorization, is an operator used in linear algebra and optimization to convert a symmetric matrix into a vector. It is designed to preserve the inner product structure between symmetric matrices and their vector representations, which is particularly useful in semidefinite programming and related fields.

Definition and convention

Given an n by n symmetric matrix A, svec(A) is a vector in R^{n(n+1)/2} formed by stacking the entries from the upper triangle (including the diagonal). A common convention is to place the diagonal elements first, followed by the off-diagonal elements with a weight of sqrt(2): svec(A) = [a_11, a_22, ..., a_nn, sqrt(2) a_12, sqrt(2) a_13, ..., sqrt(2) a_{n-1,n}]^T. The exact ordering of the off-diagonal terms can vary by source, but the sqrt(2) weighting is standard. This weighting ensures that the standard dot product of vec representations matches the trace inner product of matrices: trace(A B) = svec(A)^T svec(B) for symmetric A and B.

Relationship to other vectorizations

Svec is related to the ordinary vec operator, which stacks all columns of a matrix without weighting and yields a vector in R^{n^2}. Svec also relates to vech (which stacks only the lower (or upper) triangular part without weighting). The svec operation is often preferred in optimization when working with symmetric matrices because it preserves inner products and quadratic forms.

Applications

Svec is widely used in semidefinite programming, matrix completion, and other convex optimization problems where constraints or objective functions involve symmetric matrices. It enables efficient conversion of matrix variables and constraints into vector form suitable for standard solvers and software libraries.

Software and references

Numerous optimization toolboxes and libraries implement svec, sometimes under names like “symmetric vectorization.” It is a standard concept in optimization texts and in software such as CVX, SDPT3, SeDuMi, and MOSEK interfaces.