Determinants and their applications in mathematical physics - Vein R., Dale P..pdf

Determinants and
Their Applications in
Mathematical Physics
Robert Vein
Paul Dale
Springer
Preface
The last treatise on the theory of determinants, by T. Muir, revised and
enlarged by . 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