Use of idempotent matrices to validate linear systems software

In seeking to verify various computer implementations of Kalman filters, the extended Kalman filter (EKF), or other algorithms that rely on the same fundamental computations in a variety of different applications, it is useful to first validate software performance using low-dimensional test problems of known solution. Besides reporting the state-of-the-art in computing the critical fundamental linear system building blocks in a simple manner that is easy to understand, the author describes the easy construction of nontrivial test cases of known solution, provides a rigorous demonstration of how to augment/concatenate several low-dimensional test problems of known solution to comprise a higher-dimensional system of known solution that is also to be used in software test and validation/verification, and presents the computation of a closed-form exact optimal control solution that can also be used to validate software