Article ID Journal Published Year Pages File Type
6416542 Linear Algebra and its Applications 2013 6 Pages PDF
Abstract

Doubly nonnegative matrices arise naturally in many setting including Markov random fields (positively banded graphical models) and in the convergence analysis of Markov chains. In this short note, we settle a recent conjecture by C.R. Johnson et al. [Charles R. Johnson, Brian Lins, Olivia Walch, The critical exponent for continuous conventional powers of doubly nonnegative matrices, Linear Algebra Appl. 435 (9) (2011) 2175-2182] by proving that the critical exponent beyond which all continuous conventional powers of n-by-n doubly nonnegative matrices are doubly nonnegative is exactly n−2. We show that the conjecture follows immediately by applying a general characterization from the literature. We prove a stronger form of the conjecture by classifying all powers preserving doubly nonnegative matrices, and proceed to generalize the conjecture for broad classes of functions. We also provide different approaches for settling the original conjecture.

Related Topics
Physical Sciences and Engineering Mathematics Algebra and Number Theory
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