Complementary basic matrices (Q1827491)

This paper is a continuation of the author's paper [ibid. 373, 143--151 (2003; Zbl 1033.15010)]. The subdiagonal rank of a square matrix is defined as the order of the maximal nonsingular submatrix all of whose entries are subdiagonal, and the superdiagonal rank is defined analogously. A square matrix having both sub- and superdiagonal ranks \(\geq 1\) is called basic. In the earlier paper the author studied primarily basic matrices with LU-decomposition, while in this paper the author studies so-called complementary basic matrices without the extra condition. The main result of the earlier paper is a certain factorization of basic matrices with LU-decomposition. In this paper the author generalizes this result to complementary basic matrices and proves that the converse holds as well.