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#### Keywords

Irreducible eventually nonnegative, Strongly eventually nonnegative, Eventually reducible, Eventually r-cyclic, Cyclic index, Frobenius collection, Frobenius Jordan multiset, Jordan multiset, Jordan form

#### Abstract

A square complex matrix A is eventually nonnegative if there exists a positive integer k_0 such that for all k ≥ k_0, A^k ≥ 0; A is strongly eventually nonnegative if it is eventually nonnegative and has an irreducible nonnegative power. It is proved that a collection of elementary Jordan blocks is a Frobenius Jordan multiset with cyclic index r if and only if it is the multiset of elementary Jordan blocks of a strongly eventually nonnegative matrix with cyclic index r. A positive answer to an open question and a counterexample to a conjecture raised by Zaslavsky and Tam are given. It is also shown that for a square complex matrix A with index at most one, A is irreducible and eventually nonnegative if and only if A is strongly eventually nonnegative.

#### Recommended Citation

Hogben, Leslie; Tam, Bit-Shun; and Wilson, Ulrica.
(2015),
"Note on the Jordan form of an irreducible eventually nonnegative matrix",
*Electronic Journal of Linear Algebra*,
Volume 30.

DOI: https://doi.org/10.13001/1081-3810.3049