Basic matrices. (Q1414134)

A square matrix \(A\) over a ring is called basic if the subdiagonal and the superdiagonal ranks are at most one. The author collects a number of technical facts on basic matrices. In a complementary basic matrix the shape of the zero-nonzero entries in its inverse is the same as in its transpose. A matrix \(G\) is called elementary tridiagonal if \(G=\text{ diag}\,(I,C,I)\), where \(I\) is an identity matrix and \(C\) is a \(2\times2\) matrix. This matrix \(G\) has height \(k\) if the nonzero entries in the off-diagonals of \(C\) are in the \((n-k)\)th and \((n-k+1)\)st rows. The class of matrices of height \(k\) is denoted by \(T_k\). The author shows that a matrix is complementary basic if and only if it is a product of the form \(G_{k_1}\cdots G_{k_{n-1}}\), where \((k_1,\ldots,k_{n-1})\) is some permutation of \((1,\ldots,n-1)\) and \(G_k\in T_k\) for \(k=1,\ldots,n-1\).