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from Part I - Tools of p-adic Analysis
Published online by Cambridge University Press: 06 August 2022
We come now to the subject of metric properties of matrices over a field complete for a specified 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 of number theory. We have thus opted to first present the main points in the archimedean setting, then 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 numerical stability of certain matrix operations (including the extraction of eigenvalues). Their role in our work is similar.
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