Kronecker bases for linear matrix equations, with application to two-parameter eigenvalue problems

Abstract

AbstractThe general solutions of the homogeneous matrix equation AXCT − BXDT = 0 and the system of the matrix equations AX + BY = 0, XCT + YDT = 0 are described in terms of Kronecker canonical forms, i.e., in terms of Kronecker invariants and Kronecker bases, for pairs of matrices (A, B) and (C, D). A canonical form for a pair of commuting matrices (E, F) such that E2 = F2 = EF = 0 is discussed. These results are applied to construct a canonical basis for the second root subspace of a two-parameter eigenvalue problem. The corresponding relations for canonical invariants are given