A return panel may be rectangular, rank-deficient or have repeated singular values. The SVD still exists, so PCA compression has a valid factorisation in every case. This note uses the spectral theorem to construct the right singular vectors and links to the construction of the left vectors. We distinguish existence from uniqueness of the factor basis.
Learning objectives. By the end of this note you should be able to:
- state the full SVD existence theorem precisely, for any matrix;
- explain exactly which earlier result removes the "distinct eigenvalues" caveat;
- verify the SVD's defining relations directly;
- state what changes (or doesn't) for a matrix with repeated singular values.
In this lesson
- The full theorem
Before you start
The full lesson, with worked examples and exercise solutions, is for members.
Read the full lessonNew to SelfMadeQuant.com? See membership options or browse the Linear Algebra, Optimisation and Numerical Methods syllabus.