Studies with generalized matrix algebra

Bjerhammar, Arne

doi:10.1007/bf02531883

Cited by 9 publications

(13 citation statements)

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The classical estimation procedure according to Gauss uses the inversion of nonsingular matrices. The author has earlier presented a number of papers concerning the stochastical estimation with the use of singular inverses (Bjerhammar, 1948(Bjerhammar, -1958.The present paper gives a review of the earlier contributions and an application to selected problems. For a geodetic network all points can be considered unknown and an estimate with a generalized inverse gives the minimum variance of the observations as well as the unknowns.

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confidence: 97%

“…The classical estimation procedure according to Gauss uses the inversion of nonsingular matrices. The author has earlier presented a number of papers concerning the stochastical estimation with the use of singular inverses (Bjerhammar, 1948(Bjerhammar, -1958.…”

mentioning

confidence: 99%

“…An independent study with a generalized approach was presented a few years later (Bjerhammar, 1948(Bjerhammar, , 1949(Bjerhammar, , 1955(Bjerhammar, , 1957(Bjerhammar, , 1958. In this new presentation any matrix has at least one inverse and all singular matrices have an infinite number of inverses.…”

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Estimation with singular inverses

Bjerhammar

1971

Tellus

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The classical estimation procedure according to Gauss uses the inversion of non‐singular matrices. The author has earlier presented a number of papers concerning the stochastical estimation with the use of singular inverses (Bjerhammar, 1948–1958). The present paper gives a review of the earlier contributions and an application to selected problems. For a geodetic network all points can be considered unknown and an estimate with a generalized inverse gives the minimum variance of the observations as well as the unknowns. This estimate is invariant with respect to rotations and translations, and therefore of special interest for the study of the optimum network. It is furthermore of interest because it eliminates the dependence on the coordinate system. A number of similar applications can be found in most sciences.

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Estimation with singular inverses

Bjerhammar

1971

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“…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."…”

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“…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 Bjerhammar [2], Bose [3], Rao [9] and Rohde [13].…”

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

Pringle¹,

Rayner²

1970

SIAM Rev.

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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.

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