Lecture 18: Counting Parameters in SVD, LU, QR, Saddle Points
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 \(\lambda_1>\lambda_2>\ldots>\lambda_n\) and eigenvectors \(\boldsymbol{q}_1,\boldsymbol{q}_2,\ldots,\boldsymbol{q}_n\). Which \(i\) of those eigenvectors are a basis for an \(i\)-dimensional subspace \(Y\) with this property: The minimum of \(\boldsymbol{x}^{\mathtt{T}} S\boldsymbol{x}/\boldsymbol{x}^{\mathtt{T}} \boldsymbol{x}\) for \(\boldsymbol{x}\) in \(Y\) is \(\lambda_i\)?
10. Show that this \(2n\times 2n\) KKT matrix \(H\) has \(n\) positive and \(n\) negative eigenvalues:
$$
\begin{array}{c}\boldsymbol{S}\textbf{ positive definite}\\\boldsymbol{C}\textbf{ invertible}\end{array}\qquad \boldsymbol{H}=\left[\begin{array}{cc}\boldsymbol{S}&\boldsymbol{C}\\\boldsymbol{C}^{\mathbf{T}}&\mathbf{0}\\\end{array}\right]
$$
The first \(n\) pivots from \(S\) are positive. The last \(n\) pivots come from \(-C^{\mathtt{T}}S^{-1}C\).
