An Efficient Greedy Algorithm for finding the Nearest Simultaneous Diagonalizable Family

Abstract

Diagonalization of given multiple squared matrices by a common similarity transformation is called Simultaneous Diagonalization (SD). The approximate SD problem for finding numerically a similarity matrix which diagonalizes approximately given multiple matrices has been a long standing challenge mainly due to its nonconvexity. In this paper, we propose a new efficient greedy algorithm for finding the nearest simultaneous diagonalizable family from given matrices, and extend an elegant SD approach named the DODO algorithm to the approximate SD problem by using the proposed algorithm as its preprocessing. Numerical experiments show that the DODO algorithm preprocessed the proposed algorithm achieves more accurate estimations of solutions of the approximate SD problem than the existing ones.