Positive semidefinite matrices: Characterization via conical hulls and solution of matrix equations

Any real symmetric n??n matrix A can be described by an n(n+1)/2-component vector. Here positive-semidefiniteness of A is characterized by that vector belonging to the conical hull of a particular convex set. That characterization is used to facilitate least-squared error solution, with respect to such A, of F=AG (an equation of relevance to the design of, for example, optimization algorithms). The solution method involves finding the point in the conical hull of a convex set which is nearest to a vector. An algorithm for solving that proximal point problem is given.