The purpose of this chapter is to record a number of results that are essential tools for the discussion of the problems already described. Much of this material is elementary and is discussed here primarily to produce a consistent notation for later use. Reference will be made to some of the good available textbooks. But some of the material is given what may be an unfamiliar interpretation, and I urge everyone to at least skim the chapter.

Our basic tools are those of matrix and vector algebra as they relate to the solution of simultaneous equations, and some elementary statistical ideas mainly concerning covariance, correlation, and dispersion. Least squares is reviewed, with an emphasis placed upon the arbitrariness of the distinction between knowns, unknowns, and noise. The singular-value decomposition is a central building block, producing the clearest understanding of least squares and related formulations. I introduce the Gauss-Markov theorem and its use in making property maps, as an alternative method for obtaining solutions to simultaneous equations, and show its relation to and distinction from least squares. The chapter ends with a brief discussion of recursive least squares and estimation as essential background for the time-dependent methods of Chapter 6.

Matrix and Vector Algebra

This subject is very large and well developed, and it is not my intention to repeat material better found elsewhere (e.g., Noble & Daniel, 1977; Strang, 1988). Only a brief survey of central results is provided.