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#### Keywords

$M-$matrices, ${M_v}-$matrices, Eventually Exponentially Nonnegative Matrices, Perron-Frobenius theory

#### Abstract

${M_v}-$matrix is a matrix of the form $A = sI-B$, where $0 \le \rho (B) \le s$ and $B$ is an eventually nonnegative matrix. In this paper, $M_v-$matrices concerning the Perron-Frobenius theory are studied. Specifically, sufficient and necessary conditions for an $M_v-$matrix to have positive left and right eigenvectors corresponding to its eigenvalue with smallest real part without considering or not if $index_{0} B \leq 1$ are stated and proven. Moreover, analogous conditions for eventually nonnegative matrices or $M_v-$matrices to have all the non Perron eigenvectors or generalized eigenvectors not being nonnegative are studied. Then, equivalent properties of eventually exponentially nonnegative matrices and $M_v-$matrices are presented. Various numerical examples are given to support our theoretical findings.

abs_vol35_pp424-440.pdf (147 kB)
Abstract

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