Combinatorics Seminar
Spreaker: Samir Mondal (University of Regina)
Host: Krishnan Sivasubramanian
Title: P-matrix Powers
Time, day and date: 11:00:00 AM – 12:00:00 PM, Tuesday, July 22
Venue: Ramanuajan Hall
Abstract: A P-matrix is a matrix all of whose principal minors are positive. 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, Linear Algebra Appl., 80:159–171, 1986]: It is shown that if Ak is a P-matrix for all positive integers k, then the eigenvalues of A are positive. [In order to study (fractional and integer) powers of a matrix, we will make use of some classical aspects of the theory of matrix functions. Definition 0.1 Let f(z) be a complex function defined and analytic on a domain containing all the eigenvalues of a matrix A ∈ Mn (C). The function of A induced by f, denoted by f[A], is defined by the Cauchy integral formula
,
where Γ is any simple closed curve surrounding all eigenvalues of A and contained in the given domain. If time permits, we will explore this topic in a bit more detail.]