On the numerical ranges of matrices in max algebra

Abstract

Let $$M_{n}({\mathbb {R}}_{+})$$ M n ( R + ) be the set of all $$n \times n$$ n × n nonnegative matrices. Recently, in Tavakolipour and Shakeri (Linear Multilinear Algebra 67, 2019, https://doi.org/10.1080/03081087.2018.1478946), the concept of the numerical range in tropical algebra was introduced and an explicit formula describing it was obtained. We study the isomorphic notion of the numerical range of nonnegative matrices in max algebra and give a short proof of the known formula. Moreover, we study several generalizations of the numerical range in max algebra. Let $$1 \le k \le n$$ 1 ≤ k ≤ n be a positive integer and $$C \in M_{ n}({\mathbb {R}}_{+}).$$ C ∈ M n ( R + ) . We introduce the notions of max $$k-$$ k - numerical range and max $$C-$$ C - numerical range. Some algebraic and geometric properties of them are investigated. Also, max numerical range $$W_\text {max}(\varSigma )$$ W max ( Σ ) of a bounded set $$\varSigma$$ Σ of $$n \times n$$ n × n nonnegative matrices is introduced and some of its properties are also investigated.