Spectral properties of matrix polynomials in the max algebra

Gursoy, Buket Benek; Mason, Oliver

doi:10.1016/j.laa.2010.01.014

Cited by 7 publications

(5 citation statements)

References 11 publications

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“…The algebraic system max algebra and its isomorphic versions (max plus algebra, tropical algebra) provide an attractive way of describing a class of nonlinear problems appearing for instance in manufacturing and transportation scheduling, information technology, discrete event dynamic systems, combinatorial optimization, mathematical physics, DNA analysis, ...(see e.g. [1,3,5,6,10,11] and the references cited there). It has been used to describe these conventionally nonlinear problems in a linear fashion.…”

Section: Introductionmentioning

confidence: 99%

On the numerical ranges of matrices in max algebra

Thaghizadeh

Zahraei

Peperko

et al. 2020

Banach J. Math. Anal.

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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.

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Section: Introductionmentioning

confidence: 99%

On the numerical ranges of matrices in max algebra

Thaghizadeh

Zahraei

Peperko

et al. 2020

Banach J. Math. Anal.

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“…, m − 1}. The geometric mean of a cycle in M and the maximal cycle geometric mean µ(M ) of M are defined as similar to a simple graph; see section 1, and for more information see [12]. In the following propositions, we study the relationship between µ(C P ) and µ(A j ), where Hence, the result holds.…”

Section: Resultsmentioning

confidence: 99%

“…The following theorem, which is known as the max version of the Perron-Ferbenius theorem(see [14,Theorem 8.4.4]), is important in the context of matrices over the max algebra. At the end of this section, we study matrix polynomials in the max algebra; for more information see [12]. Suppose that…”

Section: Introductionmentioning

confidence: 99%

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Some results on matrix polynomials in the max algebra

Ghasemizadeh¹,

Aghamollaei²

2015

Banach J. Math. Anal.

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For any n × n nonnegative matrix A, and any norm . on R n , η . (A) is defined as sup { A⊗x x : x ∈ R n + , x = 0}. Let P (λ) be a matrix polynomial in the max algebra. In this paper, we introduce η . [P (λ)], as a generalization of the matrix norm η . (.), and we investigate some algebraic properties of this notion. We also study some properties of the maximum circuit geometric mean of the companion matrix of P (λ) and the relationship between this concept and the matrices P (1) and coefficients of P (λ). Some properties of η . (Ψ), for a bounded set of max matrix polynomials Ψ, are also investigated.

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“…The regular schedule is constructed by using the spectral properties over max-plus algebra. Let us remark that although max-times algebra and max-plus algebra are isomorphic, it is not straightforward to leverage the spectral properties over max-times algebra discussed in [14] to spectral properties over max-plus algebra.…”

Section: Introductionmentioning

confidence: 99%