If A is symmetric and positive definite, there is an orthogonal matrix Q for which A = Q_Λ_QT. Here Λ is the matrix of eigenvalues. Singular Value Decomposition lets us write any matrix A as a product U_Σ_VT where U and V are orthogonal and Σ is a diagonal matrix whose non-zero entries are square roots of the eigenvalues of ATA. The columns of U and V give bases for the four fundamental subspaces.
Lecture Video and Summary
- Watch the video lecture Lecture 29: Singular Value Decomposition
- Read the accompanying lecture summary (PDF)
- Lecture video transcript (PDF)
- Read Section 6.7 in the 4th edition or Section 7.1 and 7.2 in the 5th edition.
Problem Solving Video
- Watch the recitation video on Problem Solving: Computing the Singular Value Decomposition
- Recitation video transcript (PDF)
Problems and Solutions
Work the problems on your own and check your answers when you’re done.