The Matrix Pseudoinverse and Minimal Variance Estimates

This paper reviews and applies certain results concerning the matrix pseudoinverse to the general theory of estimable functions and minimal variance estimates. The paper is divided into two sections. The first section reviews and extends certain known results concerning the matrix pseudoinverse. This section is essentially non statistical. The second section uses results in the first section to state and prove a generalized version of the Gauss-Markoff Theorem concerning unbiased linear estimates having minimal variance. In the third section, an additional theorem is proven, which together with the preceding material provides a theoretical foundation for parameter estimation in orbit determination work. This foundation is then exploited to provide formulae for such parameters.