Book contents
- Frontmatter
- Contents
- Preface
- 0 Introductory remarks
- Part I Tools of p-adic Analysis
- 1 Norms on algebraic structures
- 2 Newton polygons
- 3 Ramification theory
- 4 Matrix analysis
- Part II Differential Algebra
- Part III p-adic Differential Equations on Discs and Annuli
- Part IV Difference Algebra and Frobenius Modules
- Part V Frobenius Structures
- Part VI Areas of Application
- References
- Notation
- Index
4 - Matrix analysis
Published online by Cambridge University Press: 05 August 2012
- Frontmatter
- Contents
- Preface
- 0 Introductory remarks
- Part I Tools of p-adic Analysis
- 1 Norms on algebraic structures
- 2 Newton polygons
- 3 Ramification theory
- 4 Matrix analysis
- Part II Differential Algebra
- Part III p-adic Differential Equations on Discs and Annuli
- Part IV Difference Algebra and Frobenius Modules
- Part V Frobenius Structures
- Part VI Areas of Application
- References
- Notation
- Index
Summary
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.
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- Information
- p-adic Differential Equations , pp. 55 - 74Publisher: Cambridge University PressPrint publication year: 2010