Computational geometry of positive definiteness

In matrix computations, such as in factoring matrices, Hermitian and, preferably, positive definite elements are occasionally required. Related problems can often be cast as those of existence of respective elements in a matrix subspace. For two dimensional matrix subspaces, first results in this regard are due to Finsler. To assess positive definiteness in larger dimensional cases, the task becomes computational geometric for the joint numerical range in a natural way. The Hermitian element of the Frobenius norm one with the maximal least eigenvalue is found. To this end, extreme eigenvalue computations are combined with ellipsoid and perceptron algorithms.