Two results on basic oscillatory matrices (Q1887615)

Denote by \(L_k\) (resp. \(U_k\)), \(k=1,\ldots, n-1\), the class of all \(n\times n\) matrices of the form \(D+aE_{n-k+1,n-k}\) (resp. \(D+aE_{n-k,n-k+1}\)) with a positive diagonal matrix \(D\) and a positive \(a\), where \(E_{i,j}\) is the matrix with \(1\) at the position \((i,j)\) and zeros elsewhere. A matrix \(A\) is called basic oscillatory, shortly a \(BO\)-matrix, if it admits a factoriztion \(A\in L_{i_1}L_{i_2}\cdots L_{i_{n-1}}U_{j_1}U_{j_2}\cdots U_{j_{n-1}}\), where \((i_1,i_2,\ldots,i_{n-1})\) and \((j_1,j_2,\ldots,j_{n-1})\) are permutations of \((1,2,\ldots,n-1)\). The part of the lower (upper) triangular section of a \(BO\)-matrix \(A\) that consists of zeros is a union of certain submatrices, determining the lower (upper) zig-zag shape of \(A\). The authors show the following two theorems. Let \(A\) and \(B\) be \(BO\)-matrices having the same upper zig-zag shape as well as the same lower zig-zag shape. Then their Hadamard product \(A\circ B\) is also a \(BO\)-matrix with the same upper and lower zig-zag shapes. Every oscillatory matrix \(A\) can be written in the form \(A=T_1BT_2\), where \(B\) is a \(BO\)-matrix and \(T_1\), \(T_2\) are invertible totally nonnegative matrices.