Abstract
The general solutions of the homogeneous matrix equation AXC T − BXD T = 0 and the system of the matrix equations AX + BY = 0, XC T + YD T = 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 E 2 = F 2 = 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.
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