Expressions for Generalized Inverses of a Bordered Matrix with Application to the Theory of Constrained Linear Models

Abstract

Expressions for generalized inverses of a well-known bordered matrix are derived.These expressions find application in the solution of systems of linear equations obtained by using Lagrange multipliers to find a constrained minimum. In particular, they are used to obtain explicit representations for minimum variance linear unbiased estimates of estimable linear functions in the linear model with restricted parameters.1. Preliminary results. Several types of generalized inverse matrices have been introduced in the literature and studied primarily from the point of view of utility in solving systems of linear equations. Penrose [7], for example, showed that for any matrix X (we shall, however, consider only real matrices) there exists a unique generalized inverse. This matrix, which we shall call "the generalized inverse of X" (abbreviated to "the g-inverse") and denote by Xg, is defined by the relations (1) XXX-X, XXX -X , (XXg) '-XX , (XgX)'--XX.However, as Penrose himself pointed out, in order to write a solution to a set of consistent equations Xb y (boldface symbols represent vectors) in the form b Gy, we require only the first condition, viz. XGX X. We therefore define a one-condition generalized inverse (abbreviated to g-inverse) as any matrix X g' satisfying (2) xx'x x. Such a matrix is, of course, nonunique and for convenience we write G X to stand for "G is some g-inverse of X." The algebraic properties and statistical applications of this class of g-inverses have been studied in some detail by Bjer- Rao [9] and Rohde [13].In some statistical applications two further types of generalized inverses have been found useful. These further types are defined by the first two and the first three conditions of (1). We call the former a two-condition generalized inverse (gz-inverse; Xg2) and the latter a three-condition generalized inverse (g3-inverse; Xg3).In this paper, however, we shall be primarily concerned with g-inverses, various useful properties of which are summarized in the following lemma.

Some Simplifying Results on BLUEs

“…These will be dealt with in order, with the conclusions summarized in theorems. Points (a) and (f) are well-known; they can both be found in Pringle and Rayner (1971). Point (a) can be made.…”

“…A simplified proof (presented in Section 3) shows that any estimable a'g has a BLUE m'y in which m satisfies (a) and (b). Pringle andRayner (1970, 1971) did not show how W L S E~ of g occur in the BLUE, nor the nature of the constant in the BLUE if it is of the form w'Y + a, in which w'Y is a proper random variable. Further, their formula for the variance of a BLUE incorporates Gll, which need not be unique.…”

“…They did show BLUE (A'@) = 1'8, where X'V+Xg = X'V+Y if Pz'X = 0, and they developed constrained estimates when V is singular. Pringle and Rayner (1971) Although Pringle and Raynor (1970) derived the necessary and sufficient conditions for m'Y to be a BLUE of A'@ as (a) X'm = 1 and (b) Vm = Xb, they did so through the properties of Lagrange multipliers and did not show a vector m satisfying both (a) and (b) for an arbitrary 1 in C(X'). Further, Zyskind's proof is not clear to this author.…”

“…Basic results on BLUES of estimable functions applicable to the model Y = X@ + e, in which cov (e) = V is singular, and X'X may be singular, are to be found in Albert (1972) ; Goldman and Zelen (1964) ; Kruskal (1968) ; Lowerre (1974) ; Pringle and Rayner (1971) ; Zyskind (1967); and Zyskind and Martin (1969).…”

Robust Estimation of Deformation from Observation Differences Using Some Evolutionary Optimisation Algorithms

“…The generalised inversion of matrix is determined by the bordered matrix method. Matrix is bordered by matrix , after which the generalised inversion is determined [ 54 ]: where . It is important to note that the columns of matrix are independent of the rows of the design matrix , which results in the independence of the rows of matrix from the columns of matrix .…”