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 $({\mathbb{C}}^{\ast }{)}^n$ for the matrix group $GL(n,\mathbb{C})$.

9 pages, 1 figure

References in corpus (1)

• Tropicalizing Spherical Embeddings