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Linear Equations over cones and Collatz-Wielandt numbers | | Bit-Shun Tam
; Hans Schneider
; | | Date: |
11 Sep 2001 | | Subject: | Rings and Algebras; Spectral Theory MSC-class: 15A06, 15A48 | math.RA math.SP | | Abstract: | Let $K$ be a proper cone in $IR^n$, let $A$ be an $n imes n$ real matrix that satisfies $AKsubseteq K$, let $b$ be a given vector of $K$, and let $lambda$ be a given positive real number. The following two linear equations are considered in this paper: (i)$(lambda I_n-A)x=b$, $xin K$, and (ii)$(A-lambda I_n)x=b$, $xin K$. 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 $lambda>
ho_b (A)$, and we also find a necessary condition when $lambda=
ho_b (A)$ and also when $lambda <
ho_b(A)$, sufficiently close to $
ho_b(A)$, where $
ho_b (A)$ denotes the local spectral radius of $A$ at $b$. With $lambda$ fixed, we also consider the questions of when the set $(A-lambda I_n)K igcap K$ 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. | | Source: | arXiv, math.RA/0109074 | | Services: | Forum | Review | PDF | Favorites | |
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