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Preface K. Baksalary Jerzy Zielona G&a, Poland and George P. H. Styan Mont&al, Qu&bec, Canada Almost half of the papers in this Third on Linear and Zts Applications Algebra Special Issue of Linear Algebra and Statistics were presented at the International lated Matrix Workshop on Linear Models, Experimental Designs, and ReTheory held in Tampere, Finland, 6-8 August 1990. Several papers presented at this Workshop, but which involve less linear other algebra and/or are more statistical in nature, will be published in a forthcoming Special Issue of the ]ournal of Statistical Planning and Inference. Many thanks go to Erkki Liski and Simo Puntanen, both of the Department of Mathematical Sciences/Statistics, University of Tampere, for their excellent organization of this Workshop. Two years have passed since the publication on Linear Algebra man, Friedrich years since and Vol. Rao, and and Pukelsheim, the First 82, Statistics Special October George As was noted P.H. the application and Issue (Vol. 1986, pp. Eds.). Preface on Linear matrices symmetric linear algebra definite in statistics linear models which exhibit on factor Fisher analysis, Statistics, in statistics singularities. But in multivariate six 1985; Radhakrishna Linear Algebra under a very general The next major applications properties matrices the major Perl- about can be traced back to the involving of idempotent can be found on the roots of determinantal chains of estimator model. and C. Issue D. 70, October and in deriving the distribution matrices. multidimensional Vol. Issue Special and Special Michael Eds.), Olkin, First using the concepts nonnegative 1985; to the times were in the study of Markov variables Styan, Algebra methods 1990; Ingram of the least squares and limits of their powers, of normal H. 67, June set-up which is now known as the Gauss-Markov in recent of the Second January P. 143-279; of linear algebraic work of Gauss on the optimality 127, George Styan, in the and Its Applications (Vol. forms and rank additivity impact analysis of stochastic of quadratic of the methods and inference of of from We see heavy use of linear algebra in papers scaling, and in the pioneering work of R. A. equations. LINEAR ALGEBRA AND ITS APPLlCATlONS 0 Elsevier Science Publishing Co., Inc., 1992 655 Avenue of the Americas, New York, NY 10010 176: l-2 (1992) 1 0024-3795/92/$5.00 2 JERZY Generalized matrices, inverses generalizations orderings, generalized matrices are some problems in statistics. K. BAKSALARY of matrices, separation of Chebychev projectors, AND GEORGE theorems type and Kantorovich limits of eigenvalues of the contributions to linear for singular values inequalities, stochastic of random matrices, algebra which are algebra of and Petrie motivated by The impact of linear algebra on statistics has been so substan- tial, in fact, that there are now available at least five books devoted and matrix P. H. STYAN for statistics, and a number of other entirely to linear statistical books in which linear and matrix algebra play a major role. 17 research The papers in this Third topics in linear algebra Special Issue involve the following and matrix theory and their applications to statistics and probability: diagonally range-dominant matrices, eigenvalues of matrix sums, generalized inverses (Banachiewicz-Schur form, inner, outer, symmetric reflexive), Hermite polynomials, idempotent matrices, inequalities, infinite products of matrices, iterative maximization, Laguerre polynomials, matrix commutativity, matrix derivatives, matrix norms and antinorms, matrix partial orderings (Lijwner, minus, sharp, star), numerical methods, periodicity, polynomial matrix equations, rank additivity and subtractivity of matrices, Schur complements, stochastic and V-matrices, supermultiplicativity factors, and zonal polynomials. In addition these papers cover certain aspects of the linear-algebraic and matrix-theoretic methods associated with the following topics in statistics and probability: asymptotics, canonical correlations, admissible estimators, Cochran’s analysis, theorem, covariance constrained and correlation least-squares estimators, structures, Edgeworth correspondence expansions, effi- ciency and optimality of ordinary least squares, Gauss-Markov models, generalized ridge estimators, linear models, linear regression, linear unbiased estimators, Markov chains, maximum-likelihood estimators, minimax estimamultivariate statistical analysis, tors, multicollinearity, multiple regression, nonhomogeneous Markov systems, orthogonal designs, and the Wishart distribution.