Abstract
The following three problems, are solved. When does a set of companion matrices admit simultaneous reduction to upper (lower) triangular forms? When do two sets of companion matriees admit simultaneous reduction to complementary triangular forms? When does a set of matrices admit simultaneous reduction to companion forms? The results concerning the first two problems involve (combinatorial} conditions on eigenvalues. The question of how to make these results coordinate free leads to the third problem. Its solution is phrased in terms of cyclic vectors. Simultaneous reduction to block companion forms is discussed too. The problem of complementary triangularization is related to that of complete factorization of rational matrix functions from systems theory.
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