Elements of Linear Algebra

P.M. Cohn

• Publisher: Chapman and Hall/CRC
• Year: 2017
• ISBN:
  9781351452830
• Edition: 1
• Topic: Matematica
• Subtopic: Matematica generale

This volume presents a thorough discussion of systems of linear equations and their solutions. Vectors and matrices are introduced as required and an account of determinants is given. Great emphasis has been placed on keeping the presentation as simple as possible, with many illustrative examples. While all mathematical assertions are proved, the student is led to view the mathematical content intuitively, as an aid to understanding.The text treats the coordinate geometry of lines, planes and quadrics, provides a natural application for linear algebra and at the same time furnished a geometrical interpretation to illustrate the algebraic concepts.

8

Normal forms of matrices

This chapter is devoted to the classification of square matrices. In section 8.5 we shall see that quadratic forms are described by symmetric matrices and the object will be to show that each quadratic form can be transformed to a diagonal form which is essentially unique. In terms of the matrices this provides a normal form for symmetric matrices under congruence transformations. Secondly, there is the more general problem of classifying square matrices under similarity transformations; this amounts to finding a coordinate system in which a given linear mapping has a simple description. This is superficially a different process, but it includes an important special case of the former. Here the complete answer (as well as the proof) is more complicated, so we begin by treating a number of special cases of this reduction in sections 8.1−8.4, then deal with quadratic forms in sections 8.5 and 8.6, and in section 8.7 complete the classification by presenting the Jordan normal form.

8.1 SIMILARITY OF MATRICES

Let V be a vector space of finite dimension n, say, and f:V → V a linear mapping. Relative to a basis of V with coordinates x = (x₁, …, x_n)^T we can describe f by an n × n matrix A:x⟼y, where

$y=Ax$

(8.1)

If the coordinates in a second basis of V are x′, related to x by

$x=P{x}^{\prime }$

(8.2)

with a regular matrix P, then in the new coordinates f takes the form A′:x′⟼y′, where y′ = A′x′, and hence

$A^{\prime }{x}^{\prime }=y^{\prime }=P^{-1}y=P^{-1}Ax=P^{-1}AP{x}^{\prime }$

so

$A^{\prime }=P^{-\mathrm{1...}}$

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Dedication
• Table of Contents
• Preface
• Note to the reader
• Introduction
• 2 The solution of a system of equations: the regular case
• 3 Matrices
• 4 The solution of a system of equations: the general case
• 5 Determinants
• 6 Coordinate geometry
• 7 Coordinate transformations and linear mappings
• 8 Normal forms of matrices
• 9 Applications I. Algebra and geometry
• 10 Applications II. Calculus, mechanics, economics
• Answers to the Exercises
• Notation and symbols used
• Bibliography
• Index