Page 273 - A Course in Linear Algebra with Applications
P. 273

Chapter         Eight


         EIGENVECTORS                AND       EIGENVALUES




            An   eigenvector  of  an  n  x  n  matrix  A  is  a  non-zero  n-
        column  vector  X  such that  AX  =  cX  for  some scalar  c,  which
        is called  an  eigenvalue  of  A.  Thus  the  effect  of  left  multiplica-
        tion  of an eigenvector  by  A  is merely to multiply  it  by  a scalar,
        and  when  n  <  3,  a  parallel  vector  is obtained.  Similarly,  if  T
        is a linear  operator  on  a vector  space  V,  an  eigenvector  of T  is
        a  non-zero  vector  v  of  V  such that  T(v)  =  cv  for  some  scalar
                                                                      3
        c  called  an  eigenvalue.  For  example,  if  T  is  a  rotation  in  R ,
        the  eigenvectors  of  T  are  the  non-zero  vectors  parallel  to  the
        axis  of rotation  and  the  eigenvalues  are  all  equal to  1.
             A  large  amount  of  information  about  a  matrix  or  linear
        operator  is  carried  by  its  eigenvectors  and  eigenvalues.  In
        addition,  the  theory  of  eigenvectors  and  eigenvalues  has  im-
        portant  applications  to  systems  of  linear  recurrence  relations,
        Markov  processes  and  systems  of linear  differential  equations.
        We  shall  describe  the  basic  theory  in  the  first  section  and
        then  we  give  applications  in the  following  two  sections  of  the
        chapter.


        8.1  Basic  Theory    of  Eigenvectors   and   Eigenvalues

             We  begin  with  the  fundamental  definition.  Let  A  be  an
        n  x  n  matrix  over  a  field  of  scalars  F.  An  eigenvector  of  A  is
        a  non-zero  n-column  vector  X  over  F  such  that


                                   AX  =  cX


        for  some  scalar  c  in  F;  the  scalar  c  is  then  referred  to  as  the
        eigenvalue  of  A  associated  with  the  eigenvector  X.


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