Positive Semidefinite Matrices: Characterization via Conical Hulls and Least-Squares Solution of a Matrix Equation
J.C. Allwright · SIAM Journal on Control and Optimization · 1988
Any real symmetric $n \times n$ 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.