Matrix pencil generated by a tensor product from two matrix pencils (Q2564914)

A pencil of matrices \(\lambda A-B\in\mathbb{C}[\lambda]^{m\times n}\) is regular if \(A\) and \(B\) are square matrices of the same order \(n\), and if the determinant \(|\lambda A-B|\) is not identically 0. Otherwise, the pencil is called singular. For complex \(m_i\times n_i\) matrices \(A_i\), \(B_i\) \((i=1,2)\), the authors study the pencil \(\lambda A_1\otimes A_2-B_1\otimes B_2\) arising by the tensor product from the pencils \(\lambda A_i-B_i\) \((i=1,2)\). The authors show that the regularity of the pencil \(\lambda A_1\otimes A_2-B_1\otimes B_2\) implies the regularity of the pencils \(\lambda A_i-B_i\). The converse of this theorem is false, as is shown by an example. Finally, a criterion for the regularity of \(\lambda A_1\otimes A_2-B_1\otimes B_2\) is given: The matrix pencil \(\lambda A_1\otimes A_2- B_1\otimes B_2\) is regular if and only if both pencils \(\lambda A_i-B_i\) are regular, and if either of them has infinite elementary divisors, the other one does not have the eigenvalue \(0\). Moreover, the authors derive the Weierstrass-Kronecker canonical form of the pencil \(\lambda A_1\otimes A_2- B_1\otimes B_2\) from those of the pencils \(\lambda A_i-B_i\). Some examples illustrate the results.