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## The Frobenius-Jordan form of nonnegative matrices

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Author(s) : Volker Mehrmann , Manideepa Saha

Preprint series of the Institute of Mathematics, Technische Universität Berlin

MSC 2000

15A48 Positive matrices and their generalizations; cones of matrices
15A21 Canonical forms, reductions, classification

Abstract :
In this paper we use preferred and quasi-preferred bases of generalized eigenspaces associated with the spectral radius of nonnegative matrices to analyze the existence and uniqueness of a variant of the Jordan canonical form which we call Frobenius-Jordan form. It is a combination of the classical Jordan canonical form in the part associated with the eigenvalues that are different from the spectral radius, while it is like the Frobenius normal form in the part associated with the spectral radius. Based on the Frobenius-Jordan form, spectral and combinatorial properties of nonnegative matrices are discussed. In particular, we analyze the existence of nonnegative graph representations of the generalized eigenspace associated with the spectral radius.

Keywords : Preferred basis, quasi-preferred basis, Frobenius-Jordan form