Max-plus singular values

In this paper we prove a new characterization of the max-plus singular values of a max-plus matrix, as the max-plus eigenvalues of an associated max-plus matrix pencil. This new characterization allows us to compute max-plus singu- lar values quickly and accurately. As well as capturing the asymptotic behavior of the singular values of classical matrices whose entries are exponentially pa- rameterized we show experimentally that max-plus singular values give order of magnitude approximations to the classical singular values of parameter inde- pendent classical matrices. We also discuss Hungarian scaling, which is a diagonal scaling strategy for preprocessing classical linear systems. We show that Hungarian scaling can dramatically reduce the 2-norm condition number and that this action can be explained using our new theory for max-plus singular values. Keywords: tropical algebra, max-plus algebra, singular values, diagonal scaling, condition number, optimal assignment problem We also discuss Hungarian scaling, which is a diagonal scaling strategy for preprocessing classical linear systems. We show that Hungarian scaling can dramatically reduce the d-norm condition number and that this action can be explained using our new theory for max-plus singular values.

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