The book deals with bilinear forms in real random vectors and their
generalizations as well as zonal polynomials and their applications in
handling generalized quadratic and bilinear forms. The book is mostly
self-contained. It starts from basic principles and brings the readers
to the current research level in these areas. It is developed with
detailed proofs and illustrative examples for easy readability and
self-study. Several exercises are proposed at the end of the chapters.
The complicated topic of zonal polynomials is explained in detail in
this book. The book concentrates on the theoretical developments in all
the topics covered. Some applications are pointed out but no detailed
application to any particular field is attempted. This book can be used
as a textbook for a one-semester graduate course on quadratic and
bilinear forms and/or on zonal polynomials. It is hoped that this book
will be a valuable reference source for graduate students and research
workers in the areas of mathematical statistics, quadratic and bilinear
forms and their generalizations, zonal polynomials, invariant
polynomials and related topics, and will benefit statisticians,
mathematicians and other theoretical and applied scientists who use any
of the above topics in their areas. Chapter 1 gives the preliminaries
needed in later chapters, including some Jacobians of matrix
transformations. Chapter 2 is devoted to bilinear forms in Gaussian real
ran- dom vectors, their properties, and techniques specially developed
to deal with bilinear forms where the standard methods for handling
quadratic forms become complicated.
