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Rank invariance criterion and its application to the unified theory of least squares. (English) Zbl 0694.15003
Necessary and sufficient conditions are established for the matrix product \(AB^-C\) to have its rank invariant with respect to the choice of a generalized inverse \(B^-\) of B. In particular cases, these conditions coincide with results of S. K. Mitra [Sankhyā Ser. A 34, 387-392 (1972; Zbl 0261.15008)]. The results of this paper are also discussed in the context of the unified theory of least squares introduced by C. R. Rao [ibid. 33, 371-394 (1971; Zbl 0236.62048)].
Reviewer: S.L.Campbell

MSC:
15A09 Theory of matrix inversion and generalized inverses
15A03 Vector spaces, linear dependence, rank, lineability
62J05 Linear regression; mixed models
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[1] Baksalary, J.K.; Kala, R., Range invariance of certain matrix products, Linear and multilinear algebra, 14, 89-96, (1983) · Zbl 0523.15006
[2] Baksalary, J.K.; Puntanen, S., Weighted least-squares estimation in the general Gauss-Markov model, (1988), submitted for publication
[3] Baksalary, J.K.; Puntanen, S.; Styan, G.P.H., A property of the dispersion matrix of the best linear unbiased estimator in the general Gauss-Markov model, (1988), submitted for publication
[4] Feuerverger, A.; Fraser, D.A.S., Categorical information and the singular linear model, Canad. J. statist., 8, 41-45, (1980) · Zbl 0464.62060
[5] Hartwig, R.E., 1-2 inverses and the invariance of BA+\bfc, Linear algebra appl., 11, 271-275, (1975) · Zbl 0312.15001
[6] Marsaglia, G.; Styan, G.P.H., When does rank(A + \bfb) = rank(\bfa) + rank(\bfb)?, Canad. math. bull., 15, 451-452, (1972) · Zbl 0252.15002
[7] Mitra, S.K., Fixed rank solutions of linear matrix equations, Sankhyā ser. A, 34, 387-392, (1972) · Zbl 0261.15008
[8] Mitra, S.K., Unified least squares approach to linear estimation in a general Gauss-Markov model, SIAM J. appl. math., 25, 671-680, (1973) · Zbl 0244.62054
[9] Rao, C.R., Unified theory of linear estimation, Sankhyā ser. A, 33, 371-394, (1971) · Zbl 0236.62048
[10] Rao, C.R., Corrigenda, Sankhyā ser. A, 34, 194, (1972)
[11] Rao, C.R., Corrigenda, Sankhyā ser. A, 34, 477, (1972) · Zbl 0261.62051
[12] Rao, C.R., Linear statistical inference and its applications, (1973), Wiley New York, (1st ed., 1965) · Zbl 0169.21302
[13] Rao, C.R., Unified theory of least squares, Comm. statist., 1, 1-8, (1973) · Zbl 0252.62037
[14] Rao, C.R., A unified approach to inference from linear models, (), 9-36
[15] Rao, C.R.; Mitra, S.K., Generalized inverse of matrices and its applications, (1971), Wiley New York
[16] Rao, C.R.; Mitra, S.K.; Bhimasankaram, P., Determination of a matrix by its subclasses of generalized inverses, Sankhyā ser. A, 34, 5-8, (1972) · Zbl 0274.15003
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