12. Orthogonal Matrix Factoring¶
This chapter is about orthogonal methods used to factor matrices. The first and perhaps most important step of solving a linear algebra problem is often to factor a matrix. The properties of diagonal, triangular, or unitary matrix factors can simplify how the problem is solved. Numerically stable Orthogonal matrix factoring algorithms yield unitary matrix factors. These 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, their use in factoring algorithms, and the history associated with the origins of orthogonal matrix factoring.
- 12.1. Introduction
- 12.2. The Unitary Transformation Matrices
- 12.3. The Saga of the Eigenvalue Problem
- 12.4. The Solution to the Eigenvalue Problem
- 12.4.1. The Power Method
- 12.4.2. Similarity Transformations
- 12.4.3. Direct Algorithms
- 12.4.4. Eigenvectors Following the Schur Decomposition
- 12.4.5. Iterative QR Algorithm
- 12.4.6. Schur Decomposition by the Iterative Francis Algorithm
- 12.4.7. Convergence by Similarity Transforms
- 12.4.8. Eigendecompostion from the Schur Triangular Form
- 12.5. Calculating the SVD with Unitary Transformations
