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Generalized Inverses

In: Matrix Algebra From a Statistician’s Perspective

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  • David A. Harville

    (IBM T.J. Watson Research Center, Mathematical Sciences Department)

Abstract

In statistics (and in many other disciplines), it is common to encounter linear systems whose coefficient matrices are square but not nonsingular (i.e., they are singular) and hence are not invertible (i.e., they lack inverses). For a linear system that has a nonsingular coefficient matrix, there is an intimate relationship—refer to Section 8.2a—between the solution of a linear system and the inverse of the coefficient matrix. What about a linear system whose coefficient matrix is not nonsingular? Is there a matrix comparable to the inverse (i.e., that relates to the solution of the linear system in a similar way)? The answer is yes! In fact, there are an infinite number of such matrices. These matrices, which are called generalized inverses, are the subject of this chapter. Over time, the use of generalized inverses in statistical discourse (especially that related to linear statistical models and multivariate analysis) has become increasingly routine. There is a particular generalized inverse, known as the Moore-Penrose inverse, that is sometimes singled out for special attention. Discussion of the Moore- Penrose inverse is deferred to Chapter 20. For many purposes, one generalized inverse is as good as another.

Suggested Citation

  • David A. Harville, 1997. "Generalized Inverses," Springer Books, in: Matrix Algebra From a Statistician’s Perspective, chapter 9, pages 107-132, Springer.
    • Handle: RePEc:spr:sprchp:978-0-387-22677-4_9
    DOI: 10.1007/0-387-22677-X_9
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