As a subfield of mathematics, matrix theory continues to enjoy a renais-
sance that has accelerated during the past decade, though its roots may be
traced much further back. This is due in part to stimulation from a variety
of applications and to the considerable interplay with other parts of mathe-
matics, but also to a great increase in the number and vitality of specialists
in the field. As a result, the once popular misconception that the subject has
been fully researched has been largely dispelled. The interest on the part of
the American Mathematical Society and the approximately 140 participants
in the Short Course (at the January 1989 Phoenix Meeting) on which this
volume is based is a reflection of this change. The steady growth in qual-
ity and volume of the subject's three principal journals, Linear Algebra and
its Applications, Linear and Multilinear Algebra, and the SIAM Journal on
Matrix Analysis and Applications is another. Approximately 500 different
authors have published in one of these three journals in the last two years.
Geographically, strong research centers in matrix theory have developed re-
cently in Portugal and Spain, Israel, the Netherlands, Belgium, and Hong
The purpose of the Short Course was to present a sample of the ways in
which modern matrix theory is stimulated by its interplay with other subjects.
Though the course was limited to seven speakers, the "other subjects" repre-
sented included combinatorics, probability theory, statistics, operator theory
and control theory, algebraic coding theory, partial differential equations, and
analytic function theory. Among other important examples, numerical anal-
ysis, optimization, physics and economics are, unfortunately, at most lightly
touched. There is no limit to the specific examples that might be cited.
One of the ingredients in the recent vitality of matrix theory is the va-
riety of points of view and tools brought to the subject by researchers in
different areas. This is responsible for a number of important trends in cur-
rent research. For example, the notion of majorization (mentioned in the
talk by Olkin) has become pervasive in a historically brief period of time.
The trend away from the "basis-free" point of view is illustrated by work
in combinatorial matrix theory (Brualdi, Johnson), the Hadamard product