Lecture 6: Singular Value Decomposition (SVD)
Description
Singular Value Decomposition (SVD) is the primary topic of this lecture. Professor Strang explains and illustrates how the SVD separates a matrix into rank one pieces, and that those pieces come in order of importance.
Summary
Columns of V are orthonormal eigenvectors of A_T_A.
Av = \(\sigma\)u gives orthonormal eigenvectors u of _AA_T.
\(\sigma^2 =\) eigenvalue of A_T_A = eigenvalue of _AA_T \( \neq\) 0
A = (rotation)(stretching)(rotation) \(U\Sigma\)_V_T for every A
Related section in textbook: I.8
Instructor: Prof. Gilbert Strang
Problems for Lecture 6
From textbook Section I.8
1. A symmetric matrix \(S=S^{\mathtt{T}}\) has orthonormal eigenvectors \(\boldsymbol{v}_1\) to \(\boldsymbol{v}_n\). Then any vector \(\boldsymbol{x}\) can be written as a combination \(\boldsymbol{x} = c_1\boldsymbol{v}_1 + · · · +c_n\boldsymbol{v}_n\). Explain these two formulas:
$$ \boldsymbol{x}^{\mathtt{T}}\boldsymbol{x} = {c_1}^2+ · · · +{c_n}^2 \hspace{12pt} \boldsymbol{x}^{\mathtt{T}} S\boldsymbol{x} = \lambda_1{c_1}^2+ · · · +\lambda_n{c_n}^2 $$
6. Find the σ’s and \(\boldsymbol{v}\)’s and \(\boldsymbol{u}\)’s in the SVD for \(A =\left[\begin{matrix}3 & 4\\ 0 & 5\end{matrix}\right]\). Use equation (12).
