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On scaling to an integer matrix and graphs with integer weighted cycles

Marie Maccaig 1, 2, 3
CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique, Inria Saclay - Ile de France
Abstract : Between 1970 and 1982 Hans Schneider and co-authors produced a number of results regarding matrix scalings. They demonstrated that a matrix has a diagonal similarity scaling to any matrix with entries in the subgroup generated by the cycle weights of the associated digraph, and that a matrix has an equivalent scaling to any matrix with entries related to the weights of cycles in an associated bipartite graph. Further, given matrices A and B, they produced a description of all diagonal X such that X−1AX=B. In 2005 Butkovič and Schneider used max-algebra to give a simple and efficient description of this set of scalings. In this paper we focus on the additive group of integers, and work in the max-plus algebra to give a full description of all scalings of a real matrix A to any integer matrix. We do this for four types of scalings; beginning with the familiar X−1AX, XAY and XAX scalings and finishing with a new scaling which we call a signed similarity scaling. This is a scaling of the form XAY where we specify for each row i , either xi=yi or xi=−yi. In all of our results we use necessary and sufficient conditions for existence which are based on integer weighted cycles in the associated digraph, or associated bipartite graph, of the matrix.
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Submitted on : Thursday, December 29, 2016 - 11:26:30 PM
Last modification on : Friday, April 30, 2021 - 10:00:15 AM

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Marie Maccaig. On scaling to an integer matrix and graphs with integer weighted cycles. Linear Algebra and its Applications, Elsevier, 2016, 498, pp.490--520. ⟨10.1016/j.laa.2016.01.018⟩. ⟨hal-01423517⟩



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