2270 Chapter 6 Lecture Topics S2019

1. Invertible Matrices
Invertible Matrix Theorem: All parts a to x from the textbook

2. Independence Tests and Subspaces
Independence tests: Rank, Determinant, Sampling, Wronskian, Geometry
Vector Models and Vector Spaces
3. Eigenanalysis Selected Topics
Manuscript: eigenanalysis2016-manuscript.pdf, Introduction to eigenanalysis, basic theory, AP=PD, computational algorithm, diagonalization.
Manuscript: eigenanalysisII.pdf, Discrete dynamical systems, Stochastic matrices, Eigenanalysis and footballs, Ellipse and eigenanalysis, Coordinate rotation.

4. Cayley-Hamilton Theorem
Slides: cayleyHamilton.pdf, Cayley-Hamilton Theorem, Examples
Slides: deSystemsCayleyHamilton.pdf, Cayley-Hamilton Theorem, Differential equations
Slides: orthogonality.pdf, inner product, orthogonal sets, Cauchy-Schwartz inequality, triangle inequality
Slides: fundTheoremLinearAlgebra.pdf, Orthogonal subspace W-perp, Four Fundamental subspaces, Fundamental Theorem of Linear Algebra
5. Advanced Topics in linear Algebra
Manuscript: advancedTopicsLinearAlgebra.pdf, Cayley-Hamilton proof, Jordan theory, Orthogonality and Least Squares, Gram-Schmidt, Orthonormal basis, Near Point Theorem, QR-Decomposition, Spectral Theorem, Schur's Theorem, Singular Value Decomposition, Geometry of the SVD, Four Fundamental Subspaces of Gilbert Strang
6. Jordan Form and Eigenanalysis
Manuscript: jordan-form.pdf, Jordan theory, Jordan Form, AP=PJ, Real Jordan form, map a+ib ==> 2x2 matrix, Exponential matrices.
7. Orthogonal Projections and the Near Point Theorem
Blackboard photos 3apr2019: proof that the definition of orthogonal projection does not depend on the selected orthogonal basis.