Bidiagonal decompositions of oscillating systems of vectors (Q924326)

The authors obtain necessary and sufficient conditions for a matrix \(V\) to be a matrix of eigenvectors of a totally positive matrix. Namely, this is the case if and only if \(V\) and \(V^{-T}\) are lowerly totally positive. These conditions translate into easy requirements on the parameters in the bidiagonal decompositions of \(V\) and \(V^{-T}\). By using these decompositions they give elementary proofs of the oscillating properties of \(V\). In particular, of the fact that the \(j\)th column of \(V\) has \(j-1\) changes of sign. The results include the fact that the \(Q\) matrix in a QR decomposition of a totally positive matrix belongs to the above class.