Simultaneous reduction to triangular forms after extension with zeroes
Setting out from the known Schur's lemma and \textit{N. H. McCoy}'s theorem [Bull. Am. Math. Soc. 42, 592-600 (1936; Zbl 0015.05501)] that provides a necessary and sufficient condition so that two complex matrices \(A\), \(Z\) admit simultaneous reduction to upper triangular form (that is, if there exists an invertible matrix \(S\), such that both \(S^{-1}AS\) and \(S^{-1}ZS\) are upper triangular), the paper studies when two complex matrices \(A\), \(Z\) admit simultaneous reduction to complementary triangular forms (that is, if there exists an invertible matrix \(S\), such that \(S^{-1}AS\) is upper triangular and \(S^{-1}ZS\) is lower triangular). This notion is originated from mathematical systems theory. The authors provide necessary and sufficient conditions for simultaneous reduction to complementary triangular forms in the special case of almost diagonalizable matrices, this is, matrices whose spectral polynomial is of rank of \(1\). Also, simultaneous reduction to complementary triangular forms after extension with zeros is discussed for pairs of nilpotent and Jordan matrices.
simultaneous reduction to upper triangular form
simultaneous reduction to complementary triangular forms
scientific article; zbMATH DE number 1370028
scientific article