Orthogonal Transformations

This chapter presents the orthogonal transformations (reflection and rotation) reducing the matrix A to canonical forms, which is instrumental in solving the problem of the Localization linear least squares. Reflections and rotations are computationally attractive because they are easily constructed, and they can be used to introduce zeros in a vector by properly choosing the rotation angle or the reflection plane. In this presentation, the following are presented: Brief and sufficient mathematical descriptions are given for the basic matrix reflection and rotation, Householder reflections, and Given rotations. The latter two have a key role in solving the Localization linear least-squares problem. In-place references are included, for further reading. C++ implementations, using the Eigen library, are provided.