What's in a name? To paraphrase Shakespeare's Juliet, that which - ilie
Haynsworth called the Schur complement, by any other name would be just
as beautiful. Nevertheless, her 1968 naming decision in honor of Issai
Schur (1875-1941) has gained lasting acceptance by the mathematical com-
munity. The Schur complement plays an important role in matrix analysis,
statistics, numerical analysis, and many other areas of mathematics and
its applications. Our goal is to expose the Schur complement as a rich
and basic tool in mathematical research and applications and to discuss
many significant re- sults that illustrate its power and fertility.
Although our book was originally conceived as a research reference, it
will also be useful for graduate and up- per division undergraduate
courses in mathematics, applied mathematics, and statistics. The
contributing authors have developed an exposition that makes the
material accessible to readers with a sound foundation in linear
algebra. The eight chapters of the book (Chapters 0-7) cover themes and
varia- tions on the Schur complement, including its historical
development, basic properties, eigenvalue and singular value
inequalities, matrix inequalities in both finite and infinite
dimensional settings, closure properties, and appli- cations in
statistics, probability, and numerical analysis. The chapters need not
be read in the order presented, and the reader should feel at leisure to
browse freely through topics of interest.
