Page 9 - Matrices theory and applications

Preface
viii
• Quantum chemistry is intimately related to matrix groups and their
representations.
• The case of quantum mechanics is especially interesting: Observables
are Hermitian operators, their eigenvalues are energy levels. In the
early years, quantum mechanics was called “mechanics of matrices,”
and it has now given rise to the development of the theory of large
random matrices. See [23] for a thorough account of this fashionable
topic.
This text was conceived during the years 1998–2001, on the occasion of
´
a course that I taught at the Ecole Normale Sup´erieure de Lyon. As such,
every result is accompanied by a detailed proof. During this course I tried
to investigate all the principal mathematical aspects of matrices: algebraic,
geometric, and analytic.
In some sense, this is not a specialized book. For instance, it is not as
detailed as [19] concerning numerics, or as [35] on eigenvalue problems,
or as [21] about Weyl-type inequalities. But it covers, at a slightly higher
than basic level, all these aspects, and is therefore well suited for a gradu-
ate program. Students attracted by more advanced material will find one
or two deeper results in each chapter but the first one, given with full
proofs. They will also find further information in about the half of the
170 exercises. The solutions for exercises are available on the author’s site
http://www.umpa.ens-lyon.fr/ ˜serre/exercises.pdf.
This book is organized into ten chapters. The first three contain the
basics of matrix theory and should be known by almost every graduate
student in any mathematical field. The other parts can be read more or
less independently of each other. However, exercises in a given chapter
sometimes refer to the material introduced in another one.
This text was first published in French by Masson (Paris) in 2000, under
the title Les Matrices: th´eorie et pratique. I have taken the opportunity
during the translation process to correct typos and errors, to index a list
of symbols, to rewrite some unclear paragraphs, and to add a modest
amount of material and exercises. In particular, I added three sections,
concerning alternate matrices, the singular value decomposition, and the
Moore–Penrose generalized inverse. Therefore, this edition differs from the
French one by about 10 percent of the contents.

Acknowledgments. Many thanks to the Ecole Normale Sup´erieure de Lyon
and to my colleagues who have had to put up with my talking to them
so often about matrices. Special thanks to Sylvie Benzoni for her constant
interest and useful comments.

Lyon, France Denis Serre
December 2001