淡江大學機構典藏:Item 987654321/41397
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    Please use this identifier to cite or link to this item: https://tkuir.lib.tku.edu.tw/dspace/handle/987654321/41397


    Title: Linear equations over cones and Collatz-Wielandt numbers
    Authors: Tam, Bit-Shun;Schneider, Hans
    Contributors: 淡江大學數學學系
    Keywords: Cone-preserving map;Perron–Frobenius theory;Local spectral radius;Local Perron–Schaefer condition;Nonnegative matrix;Collatz–Wielandt number;Collatz–Wielandt set;Alternating sequence
    Date: 2003-04
    Issue Date: 2013-06-13 11:24:48 (UTC+8)
    Publisher: Philadelphia: Elsevier Inc.
    Abstract: Let K be a proper cone in , let A be an n×n real matrix that satisfies AK⊆K, let b be a given vector of K, and let λ be a given positive real number. The following two linear equations are considered in this paper: (i) , and (ii) (A−λIn)x=b, x∈K. We obtain several equivalent conditions for the solvability of the first equation. For the second equation we give an equivalent condition for its solvability in case when λ>ρb(A), and we also find a necessary condition when λ=ρb(A) and also when λ<ρb(A), sufficiently close to ρb(A), where ρb(A) denotes the local spectral radius of A at b. With λ fixed, we also consider the questions of when the set (A−λIn)K∩K equals {0} or K, and what the face of K generated by the set is. Then we derive some new results about local spectral radii and Collatz–Wielandt sets (or numbers) associated with a cone-preserving map, and extend a known characterization of M-matrices among Z-matrices in terms of alternating sequences.
    Relation: Linear Algebra and Its Applications 363, pp.295-332
    DOI: 10.1016/S0024-3795(01)00565-1
    Appears in Collections:[Graduate Institute & Department of Mathematics] Journal Article

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