Two cores of a nonnegative matrix |
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Authors: | Peter Butkovi? Hans Schneider Serge? Sergeev Bit-Shun Tam |
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Institution: | 1. University of Birmingham, School of Mathematics, Edgbaston, B15 2TT, UK;2. University of Wisconsin-Madison, Department of Mathematics, 480 Lincoln Drive, Madison, WI 53706, USA;3. Tamkang University, Department of Mathematics, Tamsui, 25137, Taiwan, ROC |
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Abstract: | We prove that the sequence of eigencones (i.e., cones of nonnegative eigenvectors) of positive powers Ak of a nonnegative square matrix A is periodic both in max algebra and in nonnegative linear algebra. Using an argument of Pullman, we also show that the Minkowski sum of the eigencones of powers of A is equal to the core of A defined as the intersection of nonnegative column spans of matrix powers, also in max algebra. Based on this, we describe the set of extremal rays of the core. |
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Keywords: | Max algebra Nonnegative matrix theory Perron&ndash Frobenius theory Matrix power Eigenspace Core |
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