8. Orthogonal Matrix Factoring

This chapter is about orthogonal methods used to factor matrices. The properties of diagonal, triangular, or unitary matrix factors can simplify how a problem is solved. Numerically stable orthogonal matrix factoring algorithms trim matrices to a desired shape with unitary transformation matrices that set specified matrix elements to zero by multiplication. The factors are the trimmed matrix and products of the unitary transformation matrices, which are Givens rotation and Householder reflection matrices. This chapter describes the unitary transformation matrices and their use in factoring algorithms. Algorithms that factor matrices with the eigendecomposition and the singular value decomposition are described along with implementations. Some of the history associated with the origins of orthogonal matrix factoring and how the eigenvalue problem was solved is covered in History of the Eigenvalue Problem.

Note

The material in this chapter is reference material rather than an official part of the course. It is material that I wrote for my book. As such, the example code is in MATLAB. With your background in Python, you should not find it difficult to read and understand the MATLAB code. I can also answer any questions that you may have about the material.

Contents