Simultaneous reduction to triangular forms after extension with zeroes Original Research Article

This paper is concerned with pairs of m × m matrices A.Z for which there exists an invertible m × m matrix S such that S−1 AS is upper triangular and S1ZS is lower triangular. Such pairs are said to admit simultaneous reduction to complementary triangular forms. In particular, if such a similariy S does not exist for the pair A.Z, tollowing question is taken into consideration: Does there exist a positive integer r, such that the pair of (m + r) × (m + r) matrices image admits simultaneous reduction to complementary triangular forms, so that the pair A.Z obtains the property after extension with zeroes? It is shown that the answer to this question is mixed. An example of a pair is given which obtains the property after extension with one zero. This example is put into contrast with a large number of results that answers the question negatively, i.e., that describes situations where the pair of extended matrices admits simultaneous reduction to complementary triangular forms if and only if the original pair does.