We come now to the subject of metric properties of matrices over a field complete for a given norm. While this topic is central to our study of differential modules over nonarchimedean fields, it is based on ideas which have their origins largely outside number theory. We have thus opted to present the main points first in the archimedean setting and then to repeat the presentation for nonarchimedean fields.

The main theme is the relationship between the norms of the eigenvalues of a matrix, which are core invariants but depend on the entries of the matrix in a somewhat complicated fashion, and some less structured but more readily visible invariants. The latter are the singular values of a matrix, which play a key role in numerical linear algebra in controlling the numerical stability of certain matrix operations (including the extraction of eigenvalues). Their role in our work is similar.

Before proceeding, we set some basic notation and terminology for matrices.

Notation 4.0.1. Let Diag(σ1,…, σn) denote the n × n diagonal matrix D with Dii = σi for i = 1,…,n.

Notation 4.0.2. For A a matrix, let AT denote the transpose of A. For A an invertible square matrix, let A−T denote the inverse transpose of A.