Page 274 - A Course in Linear Algebra with Applications
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258           Chapter  Eight:  Eigenvectors  and  Eigenvalues

                 In  order to clarify  the  definition  and  illustrate  the  tech-
            nique  for  finding  eigenvectors and  eigenvalues,  an example  will
            be  worked  out  in detail.
            Example     8.1.1
            Consider  the  real  2 x 2  matrix

                                    A         - 1
                                      <1        4


            The  condition  for  the  vector
                                             'xx^
                                      M  2)
                                             x
            to  be an eigenvector  of A is that  AX  = cX   for some  scalar
            c.  This  is equivalent to (A  — cI 2)X  = 0,  which  simply  asserts
            that X is a solution  of the  linear  system

                             2 - c   - 1       Xi
                               2   4 - c       %2

            Now by  3.3.2 this linear system  will have a non-trivial  solution
            xi,  X2 if and  only  if the  determinant  of the  coefficient  matrix
            vanishes,
                                   2 - c   - 1
                                                  = 0,
                                     2   4 - <
                      2
            that  is, c  —  6c +  10 = 0.  The  roots  of this  quadratic  equa-
            tion  are  c\ = 3 + >/^T  and  C2 = 3 —   -\f—l, so these  are  the
            eigenvalues  of A.
                 The  eigenvectors  for  each  eigenvalue  are  found  by  solving
            the  linear  systems  (A  — C\I 2)X  = 0 and  (A  — c 2l2)X  =  0.  For
            example,  in the  case  of c\  we  have to solve


                           {-\-4^l)x X-                 £2=0
                                                   =
                                       2zi + (l->/ l)a;2 = 0
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