Lecture 18: Counting Parameters in SVD, LU, QR, Saddle Points

Course Info

Instructor

Prof. Gilbert Strang

Departments

Mathematics

As Taught In

Spring 2018

Level

Undergraduate

Topics

Learning Resource Types

Lecture Videos

Problem Sets

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Video Lectures

Description

In this lecture, Professor Strang reviews counting the free parameters in a variety of key matrices. He then moves on to finding saddle points from constraints and Lagrange multipliers.

Summary

Topic 1: Find $n^{2}$ parameters in $L$ and $U$ , $Q$ and $R$ , …
Find $(m + n - r) r$ parameters in a matrix of rank $r$
Topic 2: Find saddle points from constraints and Lagrange multipliers

Related section in textbook: III.2

Instructor: Prof. Gilbert Strang

Problems for Lecture 18
From textbook Section III.2

4. $S$ is a symmetric matrix with eigenvalues $λ_{1} > λ_{2} > \dots > λ_{n}$ and eigenvectors $q_{1}, q_{2}, \dots, q_{n}$ . Which $i$ of those eigenvectors are a basis for an $i$ -dimensional subspace $Y$ with this property: The minimum of $x^{T} S x / x^{T} x$ for $x$ in $Y$ is $λ_{i}$ ?