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Symmetric Matrices: Theory and Applications

Helmut Kahl

TL;DR

This work develops a unified perspective on symmetric matrices through their intimate connection with quadratic forms, extending from foundational definitions over integral domains to advanced classifications and wide-ranging applications. It presents core theorems (e.g., Witt cancellation, Hasse principle) and classical constructions (Gauss composition of binary forms) that link algebraic structure to geometric and numerical phenomena. The text then demonstrates diverse applications across numerical linear algebra (pseudoinverses, QR, Gauss–Seidel), optimization (Hessian criteria), geometry (distance and area via determinants, quadric sectors), statistics (regression loss and correlations), and cryptography (group operations in class groups). Its significance lies in providing both a deep theoretical backbone and practical tools for exploiting symmetry in matrices across multiple disciplines. Overall, the paper broadens the toolkit for analyzing symmetric matrices, enabling robust classification, efficient computation, and cross-domain applications.

Abstract

This text is a survey on symmetric matrices. It serves as a script for a module to be taught at university.

Symmetric Matrices: Theory and Applications

TL;DR

This work develops a unified perspective on symmetric matrices through their intimate connection with quadratic forms, extending from foundational definitions over integral domains to advanced classifications and wide-ranging applications. It presents core theorems (e.g., Witt cancellation, Hasse principle) and classical constructions (Gauss composition of binary forms) that link algebraic structure to geometric and numerical phenomena. The text then demonstrates diverse applications across numerical linear algebra (pseudoinverses, QR, Gauss–Seidel), optimization (Hessian criteria), geometry (distance and area via determinants, quadric sectors), statistics (regression loss and correlations), and cryptography (group operations in class groups). Its significance lies in providing both a deep theoretical backbone and practical tools for exploiting symmetry in matrices across multiple disciplines. Overall, the paper broadens the toolkit for analyzing symmetric matrices, enabling robust classification, efficient computation, and cross-domain applications.

Abstract

This text is a survey on symmetric matrices. It serves as a script for a module to be taught at university.

Paper Structure

This paper contains 26 sections, 66 theorems, 185 equations.

Table of Contents

  1. Introduction
  2. Basic notions and notations
  3. Quadratic forms
  4. Quadrics with external symmetry centre
  5. Classification
  6. Classification over the reals
  7. Classification over the rationals
  8. Classification over the integers / group structure
  9. Orthogonal group
  10. Orthogonal matrices
  11. Witt's Cancellation Theorem
  12. Automorphs of integral binary forms
  13. Linear algebraic application: linear equation systems
  14. Analytic applications
  15. Finding local extrema with the Hessian matrix
  16. ...and 11 more sections

Key Result

Proposition 2.4

For a quadratic matrix $A = (a_{i j}) \in R^{n \times n}$ the element, recursively well defined byLaplace's expansion by $j$-th column with $A_{i j}$ evolving from $A$ by deleting the $i$-th row and $j$-th column, is the same for all $j \in \mathbb{N}_n$ and equalsLaplace's expansion by $i$-th row for all $i \in \mathbb{N}_n$. In other words: for $\textnormal{adj}(A) := ((-1)^{i+j} |A_{i j}|)_{j,

Theorems & Definitions (299)

  • Definition 2.1
  • Example 1
  • Example 2
  • Definition 2.2
  • Example 3
  • Remark 2.3
  • Example 4
  • Example 5
  • Proposition 2.4
  • proof
  • ...and 289 more