Bidiagonal factorizations and quasi-oscillatory rectangular matrices
An \(m\times n\) real matrix \(A\) is called totally nonnegative if all its minors are nonnegative, and totally positive if all its minors are positive. The authors define a totally nonnegative matrix \(A\) to be quasi-oscillatory if some positive integral power of \(AA^{\top}\) is totally positive. A square matrix \(E_{k}(\alpha)\) whose diagonal entries are \(1\), whose \((k,k-1)\)st entry is \(\alpha\) and all other entries are \(0\) is called an elementary bidiagonal matrix. The authors show how to factor a totally nonnegative matrix \(A\) in terms of a diagonal matrix, reduced identity matrices and elementary bidiagonal matrices. They then give necessary and sufficient conditions on these elementary bidiagonal matrices for \(A\) to be quasi-oscillatory. They also prove that some integral power of an \(n\times n\) matrix \(A\) is totally positive if and only if \(A=BC\) where \(B\) is a lower triangular matrix, \(C\) is upper triangular, and both \(B\) and \(C\) are quasi-oscillatory.
scientific article; zbMATH DE number 5349156