Optimization: Rayleigh-Ritz and Linear Algebra

This is intended as a mini intro to some practical linear algebra.

• A detailed proof that the Rayleigh-Ritz quotient is maximized by the largest eigenvector: optimizationTutorial01.pdf
• Table of Contents:

1. Linear Algebra: Bilinear Form - A Simple Optimization Problem
   1. Definitions
   2. Inner Product
   3. Quadratic Form and the Rayleigh-Ritz Quotient
   4. A real symmetric matrix has only real eigenvalues
   5. The eigenvectors of a real symmetric matrix form an orthonormal basis
   6. Spectral decomposition of a real symmetric matrix
   7. Equality of optimization of Rayleigh-Ritz quotient
   8. 
2. Analysis
   1. Dual Space
   2. Operator Norm
   3. Sets