Contents

• Introduction
• Axiomatic Viewpoint
  • Vector space axioms
  • Order relations
  • Functions
  
  
• Equivalence Relations
  • Cartesian products
  • Functions of cartesian products
  • Symmetric bilinear functions
  • Antisymmetric multilinear functions
  • Using determinants
  
  
• Mappings Between Vector Spaces
  • Mappings
  • Consequences of symmetry
  • Canonical forms
  • Diagonal matrices
  • Commuting matrices
  • Anticommuting matrices
  • Fourier pairs
  • Equivalent matrices
  • Gerschgorin's disksGershgorin disk
  • Variational principle
  • Avoided level crossings
  • Perturbation
  • Matrices as vectors
  • Confluence
  
  
• Band Matrices
  • Band matrices
  • Sturm sequences
  • A uniform treatment for 2x2 matrices
  
  
• Applications to String Vibrations
  • Description of the physical problem
  • Solving the vibration equations
  
  
• A Variety of String Vibration Examples
  • Uniform strings
  • Joining dissimilar strings
  • Point defect
  • Diatomic string
  • Tapered string
  • Second neighbor influences
  
  
• Block Matrices
• Symmetry
  • Wave symmetry
  • Dynamical matrix symmetry
  • Groups
  • Subgroups
  • Mappings and equivalence
  • Convolutionconvolution algebra algebra
  • Matrix representation
  • Characters
  • Symmetry of a regular polygon
  
  
• Affine and Projective Algebra
  • Affine space
  • Projective space
  • Mappings of a line
  • The cross ratio
  • Fixed points for projective mappings
  
  
• Bibliography
• About this document ...