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arXiv · 1407.7059

The critical exponent for generalized doubly nonnegative matrices

Xuchen Han·Charles Johnson·Pietro Paparella
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Abstract

It is known that the critical exponent (CE) for conventional, continuous powers of $n$-by-$n$ doubly nonnegative (DN) matrices is $n-2$. Here, we consider the larger class of diagonalizable, entry-wise nonnegative $n$-by-$n$ matrices with nonnegative eigenvalues (GDN). We show that, again, a CE exists and are able to bound it with a low-coefficient quadratic. However, the CE is larger than in the DN case; in particular, 2 for $n=3$. There seems to be a connection with the index of primitivity, and a number of other observations are made and questions raised. It is shown that there is no CE for continuous Hadamard powers of GDN matrices, despite it also being $n-2$ for DN matrices.

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Xuchen Han, Charles Johnson, Pietro Paparella. 2014-07-25. The critical exponent for generalized doubly nonnegative matrices. https://doi.org/10.1080/03081087.2016.1223009

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