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arXiv · 1303.4701

The critical exponent conjecture for powers of doubly nonnegative matrices

Dominique Guillot·Apoorva Khare·Bala Rajaratnam
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Abstract

Doubly non-negative 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. [Linear Algebra Appl. 435 (2011)] 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.

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Dominique Guillot, Apoorva Khare, Bala Rajaratnam. 2013-06-26. The critical exponent conjecture for powers of doubly nonnegative matrices. https://doi.org/10.1016/j.laa.2013.06.031

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