Positive Semidefinite Matrices: Characterization via Conical Hulls and Least-Squares Solution of a Matrix Equation

Any real symmetric n × n matrix A can be described by an n(n + 1) / 2-component vector. Positive semidefiniteness of A is characterized by the associated vector belonging to the conical hull of a suitable convex set. This characterization is used to facilitate least-squared error solution, with respect to such A, of $F = AG$, where F and G are given matrices. The solution method involves finding the point in the conical hull of a convex set which is nearest to a vector. An algorithm is given for solving that proximal point problem.