Linear Algebra

427 Pages·2000·28.884 MB·English

by  A. Ramachandra Rao, P. Bhimasankaram (auth.)

TEXTS AND READINGS
IN MATHEMATICS 19
Linear Algebra
Second Edition
Texts and Readings in Mathematics
AdWory Editor
C. S. Seshadri, Chermai Mathematical Institute, Chennai.
ManagingEditor
Rajendra Bhatia, Indian Statistieallnstitute, New Delhi.
Editors
V. S. Borkar, Tata InstituteofFundamental Research, Mumbai.
R. L. Karandikar, Indian Statisticallnstitute, New Delhi.
C. Musili, University ofHyderabad, Hyderabad.
K. H. Paranjape, Institute ofMathematical Sciences, Chennai.
T. R. Ramadas, Tata Institute ofFundamental Research, Mumbai.
V. S. Sunder, InstituteofMathematieal Sciences, Chennai.
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Algebra and Functional Analysis
Linear Algebra
Second Edition
A. Ramachandra Rao
Indian Statistical Institute
Calcutta
and
P. Bhimasankaram
Indian Statistical Institute
Hydembad
HINDUSTA N
l1:lQl@.
UU U UB OOK AGENCY
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Produced from camera ready copy supplied by the Author.
ISBN 978-81-85931-26-5 ISBN 978-93-86279-01-9 (eBook)
DOI 10.1007/978-93-86279-01-9
To To
my wife my wife
Vani Vijaya
and children and children
Rajeev Chandu
Padmaja Chandana
- A.R.R. - P.B.
Preface
There have been broadly three approaches to the treatment of Lin
ear Algebra, viz., those emphasizing vector spaces, determinants and
elementary operations. The books following the first approach (for ex
ample, those by Halmos and Hoffman and Kunze) mainly use the setting
of linear transformations with a view to possible extensions of the results
to Banach and Hilbert spaces. The books following the second approach
(for example that by Mirsky) study Matrix Algebra making heavy use
of determinants. The last approach is generally used in service courses
where the emphasis is on the basic operations and computations involv
ing matrices with not much importance given for proofs and concepts.
For several applications in Science and Engineering, matrix setting
is of importance. However, the vector space approach is more suit
able for geometrie intuition leading to transparent proofs besides being
amenable for generalization to infinite-dimensional spaces. The Indian
Schoölled by Professors C. R. Raa and S. K. Mitra has successfully used
this approach to great advantage. We follow them in this book and sys·
tematically develop the elemenatry parts of matrix theory exploiting the
properties of row and column spaces of matrices. The book is meant to
be a text book at the honours level for students of Mathematics and/or
Statistics though it can be used with advantage by students of other
subjects like Physics, Computer Science, Engineering, Operational Re
search, etc. It can also serve as a reference book for scientists in various
other disciplines.
The developments in Linear Algebra during the past few decades
have brought into focus se ver al techniques like rank-factorization, gen
eralized inverses and singular value decomposition, which are hitherto
not included in elementary text books. These techniques are actually
simple enough to be taught at undergraduate (honours) level and, when
properly used, provide a better understanding of the topic besides giving
simpler proofs of results, thus enablillg the student to learn the subject
without tears. One of our aims in writing this book is to provide a
treatment incorporating these.
The book is organized in two parts. The results in Chapters 1 through
viii Preface
6 are valid ,over any field while in Chapters 7, 8 and 9 the field is assumed
to be the real field 'IR or the complex field C. We think this is very
useful because, without this separation, one may have to spend a huge
amount of time to decide whether a (frequently used) result is true over a
general field. Fields other than IR and C are very much used in subjects
like Statistics, Computer Science, Co ding Theory, Combinatorics, etc.
However, a student not interested in general fields would not lose much
by taking the field to be IR or C throughout. We assure the reader that
the proofs in Chapters 1-6 are no more complicated than they would be
if the field were assumed to be IR or C.
In an earlier version of the book, there were two chapters, one on
Linear Programming and the other on Statistical Applications, which we
have dropped for the sake of uniformity and brevity. We have, moreover,
simplified some of the proofs and the presentation of the material in
several places and included a discussion on the locus of a general equation
of the second degree in the plane.
We mention a few general features of tne book. Many of the proce
dures are given in the form of algorithms which can easily be converted
to computer programmes in any high levellanguage. However we have
not tried to give the best computationally stable algorithms as they usu
ally need a lot more theory and are available in computer packages. We
have included a large number of examples illustrating almost every defi
nition, result, procedure, etc. However, the real highlight of the book is
the large collection of exercises given at the ends of the sections. Hints
are provided along with the exercises for a few tricky ones; more hints
and solutions are given at the end of the book for the rest. We urge
each student to attempt as many exercises as possible without looking
at the solutions. However, one should not feel discouraged if he or she
needs frequent help of the solutions as there are many exercises which
are either tough or lengthy. We have marked with an asterisk exercises
and topics which, we think, are somewhat difficult alld may be omitted
in a first reading.
We had to omit some important topics like non-negative matricas
and matrix analysis mainly because of the constraint on the length of
the book. Though we have corrected the errors in the earlier version of
the book which were brought to our notice, some errors (typographical
or even conceptual) are bound to remain in spite of our best efforts.
We shall be grateful if the readers inform us of any errors found or
Preface ix
suggestions for improvement.
The list of references given at the end of the book is by no means
exhaustive and includes mainly those which have influenced us consider
ably in one form or another. More references can be obtained from some
of these.
Regarding the numbering of results: we have numbered theorems,
lemmas, definitions, examples, algorithms etc. (but not corollaries) in
the same sequence for ease in tracing them. Section 4.5 refers to the
fifth section in Chapter 4. Exercise 4.5.2 is the second exercise in Section
4.5. (4.5.2) refers to the second numbered equation (or displayed item)
in Section 4.5 and is different from Theorem 4.5.2. An index of symbols
and notations and a fairly detailed subject index are given at the end of
the book.
Though this book contains material which can be covered in just
about one year, it can also be used to teach a one-semester course as
outlined on the following page.
It is a pleasure to acknowledge the help received from various per
sons. P. Bhimasankaram is particularly grateful to Professor C. R. Rao
and Professor S. K. Mitra from whom he learnt Linear Algebra. Their
thoughts on the subject greatly influenced his own understanding of the
subject and are reflected to some extent in the present work. However,
the present authors are solely responsible for any shortcomings in this
book.
We are thankful to the authorities of the Indian Statistical Institute
for providing us the necessary facilities while writing this book. We also
thank the colleagues with whom we had discussions and the students
of several batches of B. Stat. (Hons.) and M. Stat. at the Institute for
their intelligent querries and for bearing with our experimentation using
preliminary vers ions of this book.
Finally we thank Hindustan Book Agency for their efforts in bringing
out this book in a neat form.
A. Ramachandra Rao
P. Bhimasankaram
Suggestions for a One-Semester Course
A one-semester course of about 50 lectures can be taught from the
book by covering the following portions: For shorter courses, one has to
carefully choose the topics. Here, i.j.k refers to the item i.j.k, which may
be a theorem, example or definition, etc., and i.j refers to the whole of
Section i.j excluding difficult examples or results indicated by an asterisk,
if any.
1.1-1.5,1.6.1-1.6.4,1.6.6,1.7.1-1.7.4,1.8.1, 1.8.2.
2.1-2.2,2.3.1,2.3.4,2.4-2.6, Idea ofpartitioned matrices using (2.7.2).
3.1-3.2, 3.3.1-3.3.7, 3.4.1, 3.5.1-3.5.12, 3.6.1-3.6.3, (3.7.1), 3.8.1,
3.8.2, (3.8.8), 3.9.2, 3.9.3, 3.9.7.
4.1-4.2,4.3.1,4.4.1-4.4.3,4.4.7-4.4.10,4.5.1-4.5.3.
5.1-5.3,5.5.1,5.5.2.
6.3 (assurne properties of permutations), 6.4.1-6.4.4 (omit proofs),
6.5.1,6.6.1-6.6.4,6.7.1.
7.1, 7.2.1-7.2.3, 7.2.7, 7.3.1, 7.3.8-7.3.10, 7.4.1-7.4.4, 7.4.6-7.4.9,
7.5.1-7.5.6,7.6.1-7.6.6 (excluding 7.6.3).
8.1,8.2.1-8.2.7,8.3.1,8.3.2,8.3.7,8.3.10,8.4.1, 8.4.2, 8.7.1, 8.7.2.
9.1,9.2 , 9.3.1-9.3.5,9.4.1,9.4.2,9.4.5,9.4.8,9.4.9,9.6.1.
The algorithms in the above topics may be omitted except 4.3.1,
4.4.10 and 7.4.8.