Following the results of \cite{Med}, regarding the Aluthge transform of polynomial matrices, the symboliccomputation of the Duggal transform of a polynomial matrix $A$ is developed in this paper, using the polar decomposition and the singular value decomposition of $A$. Thereat, the polynomial singular value decomposition method is utilized, which is an iterative algorithm with numerical characteristics. The introduced algorithm is proven and illustrated in numerical examples.We also represent symbolically the Duggal transform of rank-one matricesusing cross products of vectors and show that the Duggal transform of such matrices can be givenexplicitly by a closed formula and is equal to its Aluthge transform. Manuscript profile

‎In this paper we consider the minimization of a positive semidefinite quadratic form‎, ‎having a singular corresponding matrix $H$‎. ‎We state the dual formulation of the original problem and treat both problems only using the vectors $x \in \mathcal{N}(H)^\perp$ instead of the classical approach of convex optimization techniques such as the null space method‎. ‎Given this approach and based on the strong duality principle‎, ‎we provide a closed formula for the calculation of the Lagrange multipliers $\\lambda$ in the cases when (i) the constraint equation is consistent and (ii) the constraint equation is inconsistent‎, ‎using the general normal equation‎. ‎In both cases the Moore-Penrose inverse will be used to determine a unique solution of the problems‎. ‎In addition‎, ‎in the case of a consistent constraint equation‎, ‎we also give sufficient conditions for our solution to exist using the well known KKT conditions. Manuscript profile