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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.