PIMS Emergent Research Seminar: Samir Mondal

A $P$-matrix is a matrix all of whose principal minors are positive. In this talk, we demonstrate that the fractional powers of a $P$-matrix are also $P$-matrices. This insight allows us to affirmatively address a longstanding conjecture raised in [D. Hershkowitz and C.R. Johnson, Spectra of matrices with $P$-matrix powers, {\it Linear Algebra Appl.}, 80:159--171, 1986]: It is shown that if $A^k$ is a $P$-matrix for all positive integers $k$, then the eigenvalues of $A$ are positive.