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- W3043265006 abstract "Abstract Let $$M_{n}({mathbb {R}}_{+})$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:msub> <mml:mi>M</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>R</mml:mi> <mml:mo>+</mml:mo> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> be the set of all $$n times n$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>×</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:math> 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 $$1 le k le n$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>≤</mml:mo> <mml:mi>k</mml:mi> <mml:mo>≤</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:math> be a positive integer and $$C in M_{ n}({mathbb {R}}_{+}).$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>C</mml:mi> <mml:mo>∈</mml:mo> <mml:msub> <mml:mi>M</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>R</mml:mi> <mml:mo>+</mml:mo> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>.</mml:mo> </mml:mrow> </mml:math> We introduce the notions of max $$k-$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>-</mml:mo> </mml:mrow> </mml:math> numerical range and max $$C-$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>C</mml:mi> <mml:mo>-</mml:mo> </mml:mrow> </mml:math> numerical range. Some algebraic and geometric properties of them are investigated. Also, max numerical range $$W_text {max}(varSigma )$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:msub> <mml:mi>W</mml:mi> <mml:mtext>max</mml:mtext> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>Σ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> of a bounded set $$varSigma$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mi>Σ</mml:mi> </mml:math> of $$n times n$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>×</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:math> nonnegative matrices is introduced and some of its properties are also investigated." @default.
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- W3043265006 date "2020-07-18" @default.
- W3043265006 modified "2023-09-26" @default.
- W3043265006 title "On the numerical ranges of matrices in max algebra" @default.
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