This text and reference book for mathematics students and for many people working in the social sciences contains in one volume the most important properties of matrices and determinants whose elements are real or complex numbers. The theory is developed from the classical point of view of Bocher, Wedderburn, MacDuffee, and Erobernus.
Originally published in 1958.
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Yes, you can access Introduction to the Theory of Determinants and Matrices by Edward Tankard Browne in PDF and/or ePUB format, as well as other popular books in Mathematics & Algebra. We have over one million books available in our catalogue for you to explore.
of elements a, b, c, … and two rules of combination which we shall call addition (written a + b), and multiplication (written a × b, a·b, ab) such that if a and b are any two elements of
, then a + b and ab are uniquely defined elements of
. Suppose further that addition and multiplication obey the following five laws:
(1.1) a + b = b + a, (commutative law of addition);
(1.2) a + (b + c) = (a + b) + c, (associative law of addition);
(1.3) The equation a + x = b always has a solution in
;
(1.4) a(bc) = (ab)c (associative law of multiplication);
(1.5) a(b + c) = ab + ac; (b + c)a = ba + ca (distributive law).
A set of elements satisfying the above conditions is called a ring.
The condition (1.3) merely stated that subtraction is always possible in a ring. The uniqueness of subtraction is not postulated but can be proved from the conditions (1.1), …, (1.5). The unique solution of (1.3) is then written x = b – a.
If in addition to the above conditions, the relation
(1.6)
is satisfied for arbitrary elements a, b of the set, then 9i is called a commutative ring.
It can be shown that every ring
contains a unique element 0, called the zero element of the ring, which has properties that for every element a of 9t
(1.7)
(1.8)
.
If a ring
contains an element e such that ae = a for every a in
, then e is called a right unity element of the ring. Similarly, an element f such that fa = a for every a of
is called a left unity element. A ring may possess no unity element at all. On the other hand it may have right unity elements, but no left, and vice versa, (cf. ex. 10, section 5). If, however,
has both a right unity element e and a left unity element f, the two must be identical. For from the first condition fe = f, while from the second fe = e, whence e = f.
Examples of rings:
The set of all integers, positive, negative and zero;
The set of all even integers;
The set of all numbers
when a and b are integers;
The set of all rational numbers;
The set of all polynomials in a single variable with real coefficients;
The set of all continuous functions of a real variable x on the interval 0 ≤ x ≤ 1.
The set of all quaternions a + bi + cj + dk where a, b, c, d are integers and the quaternion units i, j, k sati...
Table of contents
Cover Page
Title Page
Copyright Page
Preface
Contents
Chapter I. Fundamental Concepts
Chapter II. Elementary Properties of Determinants
Chapter III. Elementary Transformations on a Matrix
Chapter IV. Further Development of the Algebra of Matrices
Chapter V. The Theory of Linear Dependence
Chapter VI. Sets of Linear Equations
Chapter VII. The Characteristic Equation of a Matrix
Chapter VIII. Special Types of Matrices
Chapter IX. Bilinear Forms
Chapter X. Quadratic Forms
Chapter XI. Real Quadratic Forms
Chapter XII. Lambda-Matrices
Chapter XIII. Equivalence of Pairs of Matrices
Chapter XIV. The Reduced Characteristic Function of a Matrix
Chapter XV. Canonical Forms of a Matrix
Chapter XVI. Scalar Polynomials in a Matrix
Chapter XVII. The Reduction of a Matrix to a Canonical Form
