Linear Systems: Consistency and Compatibility

In many instances, the solution to a problem encountered in statistics (or in some other discipline) can be reduced to a problem of solving a system of linear equations. For example, the problem of obtaining the least squares estimates of the parameters in a linear statistical model can be reduced to the problem of solving a system of linear equations called the normal equations or (if the parameters are subject to linear constraints) a system of linear equations sometimes called the constrained normal equations (e.g., Searle 1971). The focus in this chapter is on questions about the existence of (one or more) solutions to a system of linear equations—questions about the solutions themselves (when solutions exist) are deferred until Chapter 11. Such questions of existence arise in the theory of linear statistical models in determining which parametric functions are estimable (i.e., which parametric functions can be estimated from the data) and in the design of experiments (e.g., Searle 1971). The results of Sections 7.2 and 7.3 are general (i.e., applicable to any system of linear equations), while those of Section 7.4 are specific to systems of linear equations of the form of the normal equations or constrained normal equations—some of the terminology employed in discussing systems of linear equations is introduced in Section 7.1.