A MATRIX POLYNOMIAL SPECTRAL APPROACH FOR GENERAL JOINT BLOCK DIAGONALIZATION

Joint block diagonalization (JBD) of a given Hermitian matrix set {A(i)}(i=0)(p) is to find a nonsingular matrix W such that W-H A(i)W for i = 0, 1, ... , p are all block diagonal matrices with the same prescribed block diagonal structure. General JBD (GJBD) attempts to solve JBD without knowing the resulting block diagonal structure in advance, which is more difficult than JBD. In this paper, we reveal that GJBD of {A(i)}(i=0)(p) is strongly connected with the spectral information of the corresponding matrix polynomial P(lambda) = Sigma(p)(i= 0) lambda(i)A(i). Under some conditions, the solutions to GJBD are characterized by the spectral information, and a necessary and sufficient condition is given for the existence of nontrivial solutions to GJBD. In addition, based on the established theory, two feasible numerical methods are proposed to solve GJBD.NSFC [61075119, 11301013]SCI(E)EIARTICLEyfcai@math.pku.edu.cn; decaishi@gmail.com; xsf@math.pku.edu.cn2839-8633