Invariant factors as limit of singular values of a matrix

Abstract

The paper concerns a result in linear algebra motivated by ideas from tropical geometry. Let $A(t)$ be an $n \times n$ matrix whose entries are Laurent series in $t$. We show that, as $t \to 0$, logarithms of singular values of $A(t)$ approach the invariant factors of $A(t)$. This leads us to suggest logarithms of singular values of an $n \times n$ complex matrix as an analogue of the logarithm map on $(\mathbb{C}^*)^n$ for the matrix group $GL(n, \mathbb{C})$.