MATH 415 Study Guide - Final Guide: Dot Product, Principal Axis Theorem, Spectral Theorem

• School: University of Illinois
• Department: Mathematics
• Course: MATH 415
• Professor: J Li

Document Summary

Suggested practice exercises: chapter 6. 2, # 1, 2, 4, 5. Strang lecture: lecture 27: positive de nite matrices and minima. Spectral theorem: a is a symmetric matrix if at = a. e. g. 1 2 3. 3 5 5: any n n symmetric matrix a has n real eigenvalues and an orthonormal eigenbasis {q1, . If a = at , and if we have eigenvectors x, y, 1 (for 1 6= 2), then x and y must be orthogonal: = xt ay because a is symmetric! Since 1 6= 2, must have x y = 0! A similar argument shows that eigenvalues are real for symmetric (and real) a. Let f : rn r be a di erentiable function with critical point at 0. This means that all partial derivatives at 0 vanish. How to tell: look at the quadratic part of f !