Matrices with totally positive powers and their generalizations (Q2814831)

Let \(A\) be an \(n\times n\) real matrix. Then, for a given matrix property \(\mathcal{P}\), we say that \(A\) is eventually \(\mathcal{P}\) if \(A^{k}\) has property \(\mathcal{P}\) for all sufficiently large integers \(k>0\). The matrix \(A\) is positive if all entries are positive, totally positive if all minors are positive and a \(P\)-matrix if all of its principal minors are positive. In the book [Oscillation matrices and kernels and small vibrations of mechanical systems. Translation based on the 1941 Russian original. Revised ed. Providence, RI: AMS Chelsea Publishing (2002; Zbl 1002.74002)], \textit{F. R. Gantmacher} and \textit{M. G. Krein} proved a sufficient criterion for \(A^{k}\) to be strictly totally positive for some even integer \(k\). In the present paper, the author develops a theory of eventual positivity and uses this to give a necessary and sufficient condition for \(A\) to be eventually totally positive. Results on eventually \(P\)-matrices are also obtained.