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Mv−matrix is a matrix of the form A = sI−B, where 0 ≤ ρ(B) ≤ s and B is an eventually nonnegative matrix. In this paper, Mv−matrices concerning the Perron-Frobenius theory are studied. Specifically, sufficient and necessary conditions for an Mv−matrix to have positive left and right eigenvectors corresponding to its eigenvalue with smallest real part without considering or not if index0B ≤ 1 are stated and proven. Moreover, analogous conditions for eventually nonnegative matrices or Mv−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 Mv−matrices are presented. Various numerical examples are given to support our theoretical findings.

On the Perron-Frobenius Theory of $M_v-$matrices and Eventually Exponentially Nonnegative Matrices