Some perturbation results for a normalized Non-Orthogonal Joint Diagonalization problem

Non-Orthogonal Joint Diagonalization (NOJD) of a given real symmetric matrix set A={Aj}j=0p is to find a nonsingular matrix W   such that W⊤AjWW⊤AjW for j=0,1,…,pj=0,1,…,p are all as diagonal as possible. If the columns of the solution W are all required to be unit length, we call such NOJD problem as the Normalized NOJD (NNOJD) problem. In this paper, we discuss the perturbation theory for NNOJD as an optimization problem. Based on the perturbation analysis of general constrained optimization problem given in [16], we obtain an upper bound for the distance between an approximated solution of the perturbed optimal problem and the set of exact joint diagonalizers. As corollaries, a perturbation bound and an error bound are also given. Numerical examples validate the bounds.