A Discrete Approach for the Inverse Singular Value Problem in Some Quadratic Group

In this paper the solution of an inverse singular value problem is considered. First the decomposition of a real square matrix A = USigmaV is introduced, where U and V are real square matrices orthogonal with respect to a particular inner product defined through a real diagonal matrix G of order n having all the elements equal to +/-1, and Sigma is a real diagonal matrix with nonnegative elements, called G-singular values. When G is the identity matrix this decomposition is the usual SVD and Sigma is the diagonal matrix of singular values. Given a set {sigma(1),...,sigma(n)} of n real positive numbers we consider the problem to find a real matrix A having them as G-singular values. Neglecting theoretical aspects of the problem, we discuss only an algorithmic issue, trying to apply a Newton type algorithm already considered for the usual inverse singular value problem.