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20 results in Algebra Through Practice

  • Preface

  • Edited by T. S. Blyth, E. F. Robertson
  • Book:
    Algebra Through Practice
    Published online:
    05 June 2012
    Print publication:
    20 September 1984, pp vii-viii

  • Summary

    The aim of this series of problem-solvers is to provide a selection of worked examples in algebra designed to supplement undergraduate algebra courses. We have attempted, mainly with the average student in mind, to produce a varied selection of exercises while incorporating a few of a more challenging nature. Although complete solutions are included, it is intended that these should be consulted by readers only after they have attempted the questions. In this way, it is hoped that the student will gain confidence in his or her approach to the art of problem-solving which, after all, is what mathematics is all about.

    The problems, although arranged in chapters, have not been ‘graded’ within each chapter so that, if readers cannot do problem n this should not discourage them from attempting problem n + 1. A great many of the ideas involved in these problems have been used in examination papers of one sort or another. Some test papers (without solutions) are included at the end of each book; these contain questions based on the topics covered.

  • 3 - Eigenvalues and diagonalisation

  • Edited by T. S. Blyth, E. F. Robertson
  • Book:
    Algebra Through Practice
    Published online:
    05 June 2012
    Print publication:
    20 September 1984, pp 13-17

  • Summary

    In this chapter the emphasis is on the notions of eigenvalue and eigenvector of a square matrix A. These are respectively a scalar λ and a non-zero column matrix x such that Ax = λx In order to compute the λ and the x one begins by considering the system of equations (A - λIn)x = 0. These have a nonzero solution if and only if det (A - λIn) = 0. The left hand side of this equation is a polynomial of degree n which, when made monic, becomes the characteristic polynomial χA(X) of A. The zeros of the characteristic polynomial are thus the eigenvalues. It can be shown that every n × n matrix A satisfies its characteristic polynomial (the Cayley-Hamilton theorem). The minimum polynomial mA(X) of A is the monic polynomial of least degree satisfied by A. It has degree less than or equal to that of χA(A) and divides χA(X).

    If the n×n matrix A has n distinct eigenvalues λ1,…,λn and if x1,…,xn are corresponding eigenvectors then the matrix P whose ith column is xi for each i is such that P-1 exists and P-1AP = D where D = [dij] = diag {λ1,…,λn} is the diagonal matrix with dii = λi for every i. When A is real and symmetric, P can be chosen to be orthogonal (P-1 = Pt); this is achieved by normalising the eigenvectors