Difference between revisions of "Math 570: Matrix Analysis"
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Rank of a matrix and elementary matrices | Rank of a matrix and elementary matrices | ||
Spectral theory | Spectral theory | ||
| − | + | Shur's theorem | |
| − | + | Principal invariants of trace and determinant | |
| − | + | Quadratic forms and second derivative test | |
| − | + | Gerschgorin's theorem | |
Abstract vector spaces and general fields | Abstract vector spaces and general fields | ||
| − | + | Axioms | |
| − | + | Subspaces and bases | |
| − | + | Applications to general fields | |
Linear transformations | Linear transformations | ||
| − | + | Matrix of a linear transformation | |
| − | + | Rotations | |
| − | + | Eigenvalues and eigenvectors of linear transformations | |
Cannonical forms | Cannonical forms | ||
| − | + | Jordan Cannonical form | |
Markov chains and migration processes | Markov chains and migration processes | ||
| − | + | Regular Markov matrices | |
| − | + | Absorbing states and gambler's ruin | |
Inner product spaces | Inner product spaces | ||
| − | + | Gramm Schmidt process | |
| − | + | Tensor product of vectors | |
| − | + | Least squares | |
| − | + | Fredholm alternative | |
| − | + | Determinants and volume | |
Self adjoint operators | Self adjoint operators | ||
| − | + | Simultaneous diagonalization | |
| − | + | Spectral theory of self adjoint operators | |
| − | + | Positive and negative linear transformations | |
| − | + | Fractional powers | |
| − | + | Polar decompositions | |
| − | + | Applications | |
| − | + | Singular value decomposition | |
| − | + | The Frobenius norm and approximation in this norm | |
| − | + | Least squares and the Moore Penrose inverse | |
Norms for finite dimensional vector spaces | Norms for finite dimensional vector spaces | ||
| − | + | The p norms | |
| − | + | The condition number | |
| − | + | The spectral radius | |
| − | + | Sequences and series of linear operators | |
| − | + | Iterative methods for solutions of linear systems | |
=== Textbooks === | === Textbooks === | ||
Revision as of 13:23, 19 October 2010
Contents
Catalog Information
Title
Matrix Analysis.
Credit Hours
3
Prerequisite
Math 302 or 313; or equivalents.
Description
Special classes of matrices, canonical forms, matrix and vector norms, localization of eigenvalues, matrix functions, applications.
Desired Learning Outcomes
Math 570 is a one semester course on matrix analysis.
Prerequisites
Math 313 or 302 or equivalent and Math 112, 113, 314.
Minimal learning outcomes
Matrix arithmetic and Linear transformations The theory of determinants including all proofs of their properties Rank of a matrix and elementary matrices Spectral theory Shur's theorem Principal invariants of trace and determinant Quadratic forms and second derivative test Gerschgorin's theorem Abstract vector spaces and general fields Axioms Subspaces and bases Applications to general fields Linear transformations Matrix of a linear transformation Rotations Eigenvalues and eigenvectors of linear transformations Cannonical forms Jordan Cannonical form Markov chains and migration processes Regular Markov matrices Absorbing states and gambler's ruin Inner product spaces Gramm Schmidt process Tensor product of vectors Least squares Fredholm alternative Determinants and volume Self adjoint operators Simultaneous diagonalization Spectral theory of self adjoint operators Positive and negative linear transformations Fractional powers Polar decompositions Applications Singular value decomposition The Frobenius norm and approximation in this norm Least squares and the Moore Penrose inverse Norms for finite dimensional vector spaces The p norms The condition number The spectral radius Sequences and series of linear operators Iterative methods for solutions of linear systems
Textbooks
Possible textbooks for this course include (but are not limited to):
Horn and Johnson, Friedberg, Insel and Spence, or equivalent.
Additional topics
Numerical methods for finding eigenvalues Power methods The QR algorithm Rational canonical form
