Dale P., Vein R. - Determinants and Their Applications in Mathematical Physics
ISBN 0-387-98558-1 springer-Verlag
SPIN 10681036
PDF 393 pages
The last treatise on the theory of determinants, by T. Muir, revised and
enlarged by W.H. Metzler, was published by Dover Publications Inc. in
1960. It is an unabridged and corrected republication of the edition origi-nally published by Longman, Green and Co. in 1933 and contains a preface
by Metzler dated 1928. The Table of Contents of this treatise is given in
Appendix 13.
A small number of other books devoted entirely to determinants have
been published in English, but they contain little if anything of importance
that was not known to Muir and Metzler. A few have appeared in German
and Japanese. In contrast, the shelves of every mathematics library groan
under the weight of books on linear algebra, some of which contain short
chapters on determinants but usually only on those aspects of the subject
which are applicable to the chapters on matrices. There appears to be tacit
agreement among authorities on linear algebra that determinant theory is
important only as a branch of matrix theory. In sections devoted entirely
to the establishment of a determinantal relation, many authors define a
determinant by first defining a matrix M and then adding the words: “Let
det M be the determinant of the matrix M” as though determinants have
no separate existence. This belief has no basis in history. The origins of
determinants can be traced back to Leibniz (1646–1716) and their prop-erties were developed by Vandermonde (1735–1796), Laplace (1749–1827),
Cauchy (1789–1857) and Jacobi (1804–1851) whereas matrices were not in-troduced until the year of Cauchy’s death, by Cayley (1821–1895). In this
book, most determinants are defined directly.
vi Preface
It may well be perfectly legitimate to regard determinant theory as a
branch of matrix theory, but it is such a large branch and has such large
and independent roots, like a branch of a banyan tree, that it is capable
of leading an independent life. Chemistry is a branch of physics, but it
is sufficiently extensive and profound to deserve its traditional role as an
independent subject. Similarly, the theory of determinants is sufficiently
extensive and profound to justify independent study and an independent
book.
This book contains a number of features which cannot be found in any
other book. Prominent among these are the extensive applications of scaled
cofactors and column vectors and the inclusion of a large number of rela-tions containing derivatives. Older books give their readers the impression
that the theory of determinants is almost entirely algebraic in nature. If
the elements in an arbitrary determinant A are functions of a continuous
variable x, then A possesses a derivative with respect to x. The formula for
this derivative has been known for generations, but its application to the
solution of nonlinear differential equations is a recent development.
The first five chapters are purely mathematical in nature and contain old
and new proofs of several old theorems together with a number of theorems,
identities, and conjectures which have not hitherto been published. Some
theorems, both old and new, have been given two independent proofs on
the assumption that the reader will find the methods as interesting and
important as the results.
Chapter 6 is devoted to the applications of determinants in mathemat-ical physics and is a unique feature in a book for the simple reason that
these applications were almost unknown before 1970, only slowly became
known during the following few years, and did not become widely known
until about 1980. They naturally first appeared in journals on mathemat-ical physics of which the most outstanding from the determinantal point
of view is the Journal of the Physical Society of Japan. A rapid scan of
Section 15A15 in the Index of Mathematical Reviews will reveal that most
pure mathematicians appear to be unaware of or uninterested in the out-standing contributions to the theory and application of determinants made
in the course of research into problems in mathematical physics. These usu-ally appear in Section 35Q of the Index. Pure mathematicians are strongly
recommended to make themselves acquainted with these applications, for
they will undoubtedly gain inspiration from them. They will find plenty
of scope for purely analytical research and may well be able to refine the
techniques employed by mathematical physicists, prove a number of con-jectures, and advance the subject still further. Further comments on these
applications can be found in the introduction to Chapter 6.
There appears to be no general agreement on notation among writers on
determinants. We use the notion An = |aij |n and Bn = |bij |n, where i and
j are row and column parameters, respectively. The suffix n denotes the
order of the determinant and is usually reserved for that purpose. Rejecter minors of An are denoted by M( n)
ij , etc., retainer minors are denoted by
Nij , etc., simple cofactors are denoted by A( n)
ij , etc., and scaled cofactors
are denoted by Aij
n , etc. The n may be omitted from any passage if all the
determinants which appear in it have the same order. The letter D, some-times with a suffix x, t, etc., is reserved for use as a differential operator.
The letters h, i, j, k, m, p, q, r, and s are usually used as integer param-eters. The letter l is not used in order to avoid confusion with the unit
integer. Complex numbers appear in some sections and pose the problem
of conflicting priorities. The notation ω2 = −1 has been adopted since the
letters i and j are indispensable as row and column parameters, respec-tively, in passages where a large number of such parameters are required.
Matrices are seldom required, but where they are indispensable, they ap-pear in boldface symbols such as A and B with the simple convention
A = det A, B = det B, etc. The boldface symbols R and C, with suffixes,
are reserved for use as row and column vectors, respectively. Determinants,
their elements, their rejecter and retainer minors, their simple and scaled
cofactors, their row and column vectors, and their derivatives have all been
expressed in a notation which we believe is simple and clear and we wish
to see this notation adopted universally.
The Appendix consists mainly of nondeterminantal relations which have
been removed from the main text to allow the analysis to proceed without
interruption.
The Bibliography contains references not only to all the authors men-tioned in the text but also to many other contributors to the theory of
determinants and related subjects. The authors have been arranged in al-phabetical order and reference to Mathematical Reviews, Zentralblatt f¨ur
Mathematik, and Physics Abstracts have been included to enable the reader
who has no easy access to journals and books to obtain more details of their
contents than is suggested by their brief titles.
The true title of this book is The Analytic Theory of Determinants with
Applications to the Solutions of Certain Nonlinear Equations of Mathe-matical Physics, which satisfies the requirements of accuracy but lacks the
virtue of brevity. Chapter 1 begins with a brief note on Grassmann algebra
and then proceeds to define a determinant by means of a Grassmann iden-tity. Later, the Laplace expansion and a few other relations are established
by Grassmann methods. However, for those readers who find this form of
algebra too abstract for their tastes or training, classical proofs are also
given. Most of the contents of this book can be described as complicated
applications of classical algebra and differentiation.
In a book containing so many symbols, misprints are inevitable, but we
hope they are obvious and will not obstruct our readers’ progress for long.
All reports of errors will be warmly appreciated.
We are indebted to our colleague, Dr. Barry Martin, for general advice
on computers and for invaluable assistance in algebraic computing with the
Maple system on a Macintosh computer, especially in the expansion and
factorization of determinants. We are also indebted by Lynn Burton for
the most excellent construction and typing of a complicated manuscript in
Microsoft Word programming language Formula on a Macintosh computer
in camera-ready form.
Birmingham, U.K. P.R. Vein
P. Dale
SPIN 10681036
PDF 393 pages
The last treatise on the theory of determinants, by T. Muir, revised and
enlarged by W.H. Metzler, was published by Dover Publications Inc. in
1960. It is an unabridged and corrected republication of the edition origi-nally published by Longman, Green and Co. in 1933 and contains a preface
by Metzler dated 1928. The Table of Contents of this treatise is given in
Appendix 13.
A small number of other books devoted entirely to determinants have
been published in English, but they contain little if anything of importance
that was not known to Muir and Metzler. A few have appeared in German
and Japanese. In contrast, the shelves of every mathematics library groan
under the weight of books on linear algebra, some of which contain short
chapters on determinants but usually only on those aspects of the subject
which are applicable to the chapters on matrices. There appears to be tacit
agreement among authorities on linear algebra that determinant theory is
important only as a branch of matrix theory. In sections devoted entirely
to the establishment of a determinantal relation, many authors define a
determinant by first defining a matrix M and then adding the words: “Let
det M be the determinant of the matrix M” as though determinants have
no separate existence. This belief has no basis in history. The origins of
determinants can be traced back to Leibniz (1646–1716) and their prop-erties were developed by Vandermonde (1735–1796), Laplace (1749–1827),
Cauchy (1789–1857) and Jacobi (1804–1851) whereas matrices were not in-troduced until the year of Cauchy’s death, by Cayley (1821–1895). In this
book, most determinants are defined directly.
vi Preface
It may well be perfectly legitimate to regard determinant theory as a
branch of matrix theory, but it is such a large branch and has such large
and independent roots, like a branch of a banyan tree, that it is capable
of leading an independent life. Chemistry is a branch of physics, but it
is sufficiently extensive and profound to deserve its traditional role as an
independent subject. Similarly, the theory of determinants is sufficiently
extensive and profound to justify independent study and an independent
book.
This book contains a number of features which cannot be found in any
other book. Prominent among these are the extensive applications of scaled
cofactors and column vectors and the inclusion of a large number of rela-tions containing derivatives. Older books give their readers the impression
that the theory of determinants is almost entirely algebraic in nature. If
the elements in an arbitrary determinant A are functions of a continuous
variable x, then A possesses a derivative with respect to x. The formula for
this derivative has been known for generations, but its application to the
solution of nonlinear differential equations is a recent development.
The first five chapters are purely mathematical in nature and contain old
and new proofs of several old theorems together with a number of theorems,
identities, and conjectures which have not hitherto been published. Some
theorems, both old and new, have been given two independent proofs on
the assumption that the reader will find the methods as interesting and
important as the results.
Chapter 6 is devoted to the applications of determinants in mathemat-ical physics and is a unique feature in a book for the simple reason that
these applications were almost unknown before 1970, only slowly became
known during the following few years, and did not become widely known
until about 1980. They naturally first appeared in journals on mathemat-ical physics of which the most outstanding from the determinantal point
of view is the Journal of the Physical Society of Japan. A rapid scan of
Section 15A15 in the Index of Mathematical Reviews will reveal that most
pure mathematicians appear to be unaware of or uninterested in the out-standing contributions to the theory and application of determinants made
in the course of research into problems in mathematical physics. These usu-ally appear in Section 35Q of the Index. Pure mathematicians are strongly
recommended to make themselves acquainted with these applications, for
they will undoubtedly gain inspiration from them. They will find plenty
of scope for purely analytical research and may well be able to refine the
techniques employed by mathematical physicists, prove a number of con-jectures, and advance the subject still further. Further comments on these
applications can be found in the introduction to Chapter 6.
There appears to be no general agreement on notation among writers on
determinants. We use the notion An = |aij |n and Bn = |bij |n, where i and
j are row and column parameters, respectively. The suffix n denotes the
order of the determinant and is usually reserved for that purpose. Rejecter minors of An are denoted by M( n)
ij , etc., retainer minors are denoted by
Nij , etc., simple cofactors are denoted by A( n)
ij , etc., and scaled cofactors
are denoted by Aij
n , etc. The n may be omitted from any passage if all the
determinants which appear in it have the same order. The letter D, some-times with a suffix x, t, etc., is reserved for use as a differential operator.
The letters h, i, j, k, m, p, q, r, and s are usually used as integer param-eters. The letter l is not used in order to avoid confusion with the unit
integer. Complex numbers appear in some sections and pose the problem
of conflicting priorities. The notation ω2 = −1 has been adopted since the
letters i and j are indispensable as row and column parameters, respec-tively, in passages where a large number of such parameters are required.
Matrices are seldom required, but where they are indispensable, they ap-pear in boldface symbols such as A and B with the simple convention
A = det A, B = det B, etc. The boldface symbols R and C, with suffixes,
are reserved for use as row and column vectors, respectively. Determinants,
their elements, their rejecter and retainer minors, their simple and scaled
cofactors, their row and column vectors, and their derivatives have all been
expressed in a notation which we believe is simple and clear and we wish
to see this notation adopted universally.
The Appendix consists mainly of nondeterminantal relations which have
been removed from the main text to allow the analysis to proceed without
interruption.
The Bibliography contains references not only to all the authors men-tioned in the text but also to many other contributors to the theory of
determinants and related subjects. The authors have been arranged in al-phabetical order and reference to Mathematical Reviews, Zentralblatt f¨ur
Mathematik, and Physics Abstracts have been included to enable the reader
who has no easy access to journals and books to obtain more details of their
contents than is suggested by their brief titles.
The true title of this book is The Analytic Theory of Determinants with
Applications to the Solutions of Certain Nonlinear Equations of Mathe-matical Physics, which satisfies the requirements of accuracy but lacks the
virtue of brevity. Chapter 1 begins with a brief note on Grassmann algebra
and then proceeds to define a determinant by means of a Grassmann iden-tity. Later, the Laplace expansion and a few other relations are established
by Grassmann methods. However, for those readers who find this form of
algebra too abstract for their tastes or training, classical proofs are also
given. Most of the contents of this book can be described as complicated
applications of classical algebra and differentiation.
In a book containing so many symbols, misprints are inevitable, but we
hope they are obvious and will not obstruct our readers’ progress for long.
All reports of errors will be warmly appreciated.
We are indebted to our colleague, Dr. Barry Martin, for general advice
on computers and for invaluable assistance in algebraic computing with the
Maple system on a Macintosh computer, especially in the expansion and
factorization of determinants. We are also indebted by Lynn Burton for
the most excellent construction and typing of a complicated manuscript in
Microsoft Word programming language Formula on a Macintosh computer
in camera-ready form.
Birmingham, U.K. P.R. Vein
P. Dale
