Sylvester matrix equation for matrix pencils
Let be given complex matrices \(B_1\), \(A_1 \in \mathbb{C}^{p_1 \times q_1}\) and \(B_2\), \(A_2 \in \mathbb{C}^{p_2 \times q_2}\), and consider the homogeneous system of equations \[ A_1 X - YA_2 = 0, \quad B_1 X - YB_2 = 0. \tag{1} \] The dimension of the set of solutions of (1) is obtained in terms of the Kronecker invariants of the matrix pencils \(\lambda B_1 - A_1\) and \(\lambda B_2 - A_2\). Necessary and sufficients conditions for strict equivalence of matrix pencils are given in terms of ranks of several matrices, in particular, equality of ranks of certain \(2 \times 2\) block matrices, where each block is a Kronecker product. These conditions represent a generalization of the matrix similarity criterion given by \textit{S. Friedland} [Lect. Appl. Math. 18, AMS, 43-85 (1980; MR 82b:15014)]. Analogous necessary and sufficient conditions are given for existence of solutions of the inhomogeneous system \(A_1 X - YA_2 = A_3\), \(B_1X - YB_2 = B_3\).
scientific article; zbMATH DE number 841760
scientific article