reduce-historical/r34/xlog/matrix.log

Sat Jun 29 13:44:52 PDT 1991
REDUCE 3.4, 15-Jul-91 ...

1: 1: 
2: 2: 
3: 3: % Tests of eigenfunction/eigenvalue code.

v := mat((1,1,-1,1,0),(1,2,-1,0,1),(-1,2,3,-1,0),
        (1,-2,1,2,-1),(2,1,-1,3,0))$



mateigen(v,et);


    4       3        2
{{ET  - 6*ET  + 13*ET  + 5*ET - 5,

  1,


  [       5*ARBCOMPLEX(1)*(ET - 2)       ]
  [     ----------------------------     ]
  [          3        2                  ]
  [      2*ET  - 10*ET  + 23*ET + 5      ]
  [                                      ]
  [                        2             ]
  [ ARBCOMPLEX(1)*ET*( - ET  + 6*ET - 8) ]
  [--------------------------------------]
  [          3        2                  ]
  [      2*ET  - 10*ET  + 23*ET + 5      ]
  [                                      ]
  [    ARBCOMPLEX(1)*ET*( - 3*ET + 7)    ]
  [   --------------------------------   ]
  [          3        2                  ]
  [      2*ET  - 10*ET  + 23*ET + 5      ]
  [                                      ]
  [                    3       2         ]
  [   ARBCOMPLEX(1)*(ET  - 4*ET  + 10)   ]
  [  ----------------------------------  ]
  [          3        2                  ]
  [      2*ET  - 10*ET  + 23*ET + 5      ]
  [                                      ]
  [            ARBCOMPLEX(1)             ]

  },

 {ET - 2,

  1,


  [      0      ]
  [             ]
  [      0      ]
  [             ]
  [ARBCOMPLEX(2)]
  [             ]
  [ARBCOMPLEX(2)]
  [             ]
  [ARBCOMPLEX(2)]

  }}


eigv := third first ws$



% Now check if the equation for the eigenvectors is fulfilled.  Note
% that also the last component is zero due to the eigenvalue equation.
v*eigv-et*eigv;


[                          0                           ]
[                                                      ]
[                    4       3        2                ]
[   ARBCOMPLEX(1)*(ET  - 6*ET  + 13*ET  + 5*ET - 5)    ]
[  -------------------------------------------------   ]
[                 3        2                           ]
[             2*ET  - 10*ET  + 23*ET + 5               ]
[                                                      ]
[                          0                           ]
[                                                      ]
[                      4       3        2              ]
[  ARBCOMPLEX(1)*( - ET  + 6*ET  - 13*ET  - 5*ET + 5)  ]
[ ---------------------------------------------------- ]
[                  3        2                          ]
[              2*ET  - 10*ET  + 23*ET + 5              ]
[                                                      ]
[                       4       3        2             ]
[ 2*ARBCOMPLEX(1)*( - ET  + 6*ET  - 13*ET  - 5*ET + 5) ]
[------------------------------------------------------]
[                  3        2                          ]
[              2*ET  - 10*ET  + 23*ET + 5              ]



% Example of degenerate eigenvalues.

u := mat((2,-1,1),(0,1,1),(-1,1,1))$



mateigen(u,eta);


{{ETA - 1,2,

  [ARBCOMPLEX(3)]
  [             ]
  [ARBCOMPLEX(3)]
  [             ]
  [      0      ]

  },

 {ETA - 2,1,

  [      0      ]
  [             ]
  [ARBCOMPLEX(4)]
  [             ]
  [ARBCOMPLEX(4)]

  }}



% Example of a fourfold degenerate eigenvalue with two corresponding
% eigenvectors.

w := mat((1,-1,1,-1),(-3,3,-5,4),(8,-4,3,-4),
         (15,-10,11,-11))$



mateigen(w,al);


{{AL + 1,

  4,


  [            ARBCOMPLEX(5)             ]
  [           ---------------            ]
  [                  5                   ]
  [                                      ]
  [  - 5*ARBCOMPLEX(6) + 7*ARBCOMPLEX(5) ]
  [--------------------------------------]
  [                  5                   ]
  [                                      ]
  [            ARBCOMPLEX(5)             ]
  [                                      ]
  [            ARBCOMPLEX(6)             ]

  }}


eigw := third first ws;


        [            ARBCOMPLEX(5)             ]
        [           ---------------            ]
        [                  5                   ]
        [                                      ]
        [  - 5*ARBCOMPLEX(6) + 7*ARBCOMPLEX(5) ]
EIGW := [--------------------------------------]
        [                  5                   ]
        [                                      ]
        [            ARBCOMPLEX(5)             ]
        [                                      ]
        [            ARBCOMPLEX(6)             ]



w*eigw - al*eigw;


       - ARBCOMPLEX(5)*(AL + 1)
MAT((---------------------------),
                  5

    ((5*ARBCOMPLEX(6)*AL + 5*ARBCOMPLEX(6) - 7*ARBCOMPLEX(5)*AL

       - 7*ARBCOMPLEX(5))/5),

    ( - ARBCOMPLEX(5)*(AL + 1)),

    ( - ARBCOMPLEX(6)*(AL + 1)))



% Calculate the eigenvectors and eigenvalue equation.

f := mat((0,ex,ey,ez),(-ex,0,bz,-by),(-ey,-bz,0,bx),
        (-ez,by,-bx,0))$



factor om;



mateigen(f,om);


    4     2    2     2     2     2     2     2      2   2
{{OM  + OM *(EX  + EY  + EZ  + BZ  + BY  + BX ) + EX *BX

                                       2   2
   + 2*EX*EY*BY*BX + 2*EX*EZ*BZ*BX + EY *BY  + 2*EY*EZ*BZ*BY

       2   2
   + EZ *BZ ,

  1,


          2
  MAT(((OM *ARBCOMPLEX(7)*EZ + OM*ARBCOMPLEX(7)*( - EX*BY + EY*BX)

                                                        3
         + ARBCOMPLEX(7)*BZ*(EX*BX + EY*BY + EZ*BZ))/(OM

                   2     2     2
           + OM*(EX  + EY  + BZ ))),

             2
      (( - OM *ARBCOMPLEX(7)*BY + OM*ARBCOMPLEX(7)*( - EX*EZ + BZ*BX)

                                                          3
         - (ARBCOMPLEX(7)*EY)*(EX*BX + EY*BY + EZ*BZ))/(OM

                   2     2     2
           + OM*(EX  + EY  + BZ ))),

          2
      ((OM *ARBCOMPLEX(7)*BX + OM*ARBCOMPLEX(7)*( - EY*EZ + BZ*BY)

                                                        3
         + ARBCOMPLEX(7)*EX*(EX*BX + EY*BY + EZ*BZ))/(OM

                   2     2     2
           + OM*(EX  + EY  + BZ ))),

      (ARBCOMPLEX(7)))

}}


% Specialize to perpendicular electric and magnetic field.

let ez=0,ex=0,by=0;



% Note that we find two eigenvectors to the double eigenvalue 0
% (as it must be).

mateigen(f,om);


{{OM,

  2,


  [ ARBCOMPLEX(9)*BX - ARBCOMPLEX(8)*BZ ]
  [-------------------------------------]
  [                 EY                  ]
  [                                     ]
  [            ARBCOMPLEX(8)            ]
  [                                     ]
  [                  0                  ]
  [                                     ]
  [            ARBCOMPLEX(9)            ]

  },

    2     2     2     2
 {OM  + EY  + BZ  + BX ,

  1,


  [        - ARBCOMPLEX(10)*EY       ]
  [      ----------------------      ]
  [                BX                ]
  [                                  ]
  [        - ARBCOMPLEX(10)*BZ       ]
  [      ----------------------      ]
  [                BX                ]
  [                                  ]
  [                   2     2     2  ]
  [ ARBCOMPLEX(10)*(EY  + BZ  + BX ) ]
  [----------------------------------]
  [              OM*BX               ]
  [                                  ]
  [          ARBCOMPLEX(10)          ]

  }}


% The following has 1 as a double eigenvalue. The corresponding
% eigenvector must involve two arbitrary constants.

j := mat((9/8,1/4,-sqrt(3)/8),
         (1/4,3/2,-sqrt(3)/4),
         (-sqrt(3)/8,-sqrt(3)/4,11/8));


     [     9             1          - SQRT(3) ]
     [    ---           ---       ------------]
     [     8             4             8      ]
     [                                        ]
     [     1             3          - SQRT(3) ]
J := [    ---           ---       ------------]
     [     4             2             4      ]
     [                                        ]
     [  - SQRT(3)     - SQRT(3)        11     ]
     [------------  ------------      ----    ]
     [     8             4             8      ]



mateigen(j,x);


{{X - 1,

  2,


  [SQRT(3)*ARBCOMPLEX(12) - 2*ARBCOMPLEX(11)]
  [                                         ]
  [             ARBCOMPLEX(11)              ]
  [                                         ]
  [             ARBCOMPLEX(12)              ]

  },

 {X - 2,

  1,


  [   - SQRT(3)*ARBCOMPLEX(13)  ]
  [ --------------------------- ]
  [              3              ]
  [                             ]
  [  - 2*SQRT(3)*ARBCOMPLEX(13) ]
  [-----------------------------]
  [              3              ]
  [                             ]
  [       ARBCOMPLEX(13)        ]

  }}


% The following is a good consistency check.

sym := mat(
   (0,  1/2,  1/(2*sqrt(2)),  0,  0),
   (1/2,  0,  1/(2*sqrt(2)),  0,  0),
   (1/(2*sqrt(2)),  1/(2*sqrt(2)),  0,  1/2,  1/2),
   (0,  0,  1/2,  0,  0),
   (0,  0,  1/2,  0,  0))$



ans := mateigen(sym,eta);


ANS := {{ETA,

         1,


         [        0        ]
         [                 ]
         [        0        ]
         [                 ]
         [        0        ]
         [                 ]
         [ - ARBCOMPLEX(14)]
         [                 ]
         [ ARBCOMPLEX(14)  ]

         },

        {ETA - 1,

         1,


         [ 2*ARBCOMPLEX(15) ]
         [------------------]
         [     SQRT(2)      ]
         [                  ]
         [ 2*ARBCOMPLEX(15) ]
         [------------------]
         [     SQRT(2)      ]
         [                  ]
         [ 2*ARBCOMPLEX(15) ]
         [                  ]
         [  ARBCOMPLEX(15)  ]
         [                  ]
         [  ARBCOMPLEX(15)  ]

         },

        {2*ETA + 1,

         1,


         [ - ARBCOMPLEX(16)]
         [                 ]
         [ ARBCOMPLEX(16)  ]
         [                 ]
         [        0        ]
         [                 ]
         [        0        ]
         [                 ]
         [        0        ]

         },

              2
        {4*ETA  + 2*ETA - 1,

         1,


         [        - ARBCOMPLEX(17)       ]
         [      -------------------      ]
         [         2*SQRT(2)*ETA         ]
         [                               ]
         [        - ARBCOMPLEX(17)       ]
         [      -------------------      ]
         [         2*SQRT(2)*ETA         ]
         [                               ]
         [ ARBCOMPLEX(17)*( - 2*ETA + 1) ]
         [-------------------------------]
         [             2*ETA             ]
         [                               ]
         [        ARBCOMPLEX(17)         ]
         [                               ]
         [        ARBCOMPLEX(17)         ]

         }}


% Check of correctness for this example.

for each j in ans do
  for each k in solve(first j,eta) do
      write sub(k,sym*third j - eta*third j);


[0]
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[0]
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[0]
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[0]
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[0]


[0]
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[0]
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[0]
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[0]
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[0]


[0]
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[0]
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[0]


[0]
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% Tests of nullspace operator.

a1 := mat((1,2,3,4),(5,6,7,8));


      [1  2  3  4]
A1 := [          ]
      [5  6  7  8]



nullspace a1;


{

 [ 1  ]
 [    ]
 [ 0  ]
 [    ]
 [ - 3]
 [    ]
 [ 2  ]

 ,

 [ 0  ]
 [    ]
 [ 1  ]
 [    ]
 [ - 2]
 [    ]
 [ 1  ]

 }

 
b1 := {{1,2,3,4},{5,6,7,8}};


B1 := {{1,2,3,4},{5,6,7,8}}


nullspace b1;


{{1,0,-3,2},{0,1,-2,1}}


% Example taken from a bug report for another CA system.

c1 :=
{{(p1**2*(p1**2 + p2**2 + p3**2 - s*z - z**2))/(p1**2 + p3**2), 0,
   (p1*p3*(p1**2 + p2**2 + p3**2 - s*z - z**2))/(p1**2 + p3**2),
   -((p1**2*p2*(s + z))/(p1**2 + p3**2)), p1*(s + z),
   -((p1*p2*p3*(s + z))/(p1**2 + p3**2)),
   -((p1*p3*(p1**2 + p2**2 + p3**2))/(p1**2 + p3**2)), 0,
   (p1**2*(p1**2 + p2**2 + p3**2))/(p1**2 + p3**2)},
   {0, 0, 0, 0, 0, 0, 0, 0, 0},
  {(p1*p3*(p1**2 + p2**2 + p3**2 - s*z - z**2))/(p1**2 + p3**2), 0,
   (p3**2*(p1**2 + p2**2 + p3**2 - s*z - z**2))/(p1**2 + p3**2),
   -((p1*p2*p3*(s + z))/(p1**2 + p3**2)), p3*(s + z),
   -((p2*p3**2*(s + z))/(p1**2 + p3**2)),
   -((p3**2*(p1**2 + p2**2 + p3**2))/(p1**2 + p3**2)), 0,
   (p1*p3*(p1**2 + p2**2 + p3**2))/(p1**2 + p3**2)},
  {-((p1**2*p2*(s + z))/(p1**2 + p3**2)), 0,
   -((p1*p2*p3*(s + z))/(p1**2 + p3**2)),
   -((p1**2*p2**2*(s + 2*z))/((p1**2 + p3**2)*z)), (p1*p2*(s + 2*z))/z,
   -((p1*p2**2*p3*(s + 2*z))/((p1**2 + p3**2)*z)),
   -((p1*p2*p3*z)/(p1**2 + p3**2)), 0, (p1**2*p2*z)/(p1**2 + p3**2)},
  {p1*(s + z), 0, p3*(s + z), (p1*p2*(s + 2*z))/z,
   -(((p1**2+p3**2)*(s+ 2*z))/z), (p2*p3*(s + 2*z))/z, p3*z,0, -(p1*z)},
  {-((p1*p2*p3*(s + z))/(p1**2 + p3**2)), 0,
   -((p2*p3**2*(s + z))/(p1**2 + p3**2)),
   -((p1*p2**2*p3*(s + 2*z))/((p1**2 + p3**2)*z)), (p2*p3*(s + 2*z))/z,
   -((p2**2*p3**2*(s + 2*z))/((p1**2 + p3**2)*z)),
   -((p2*p3**2*z)/(p1**2 + p3**2)), 0, (p1*p2*p3*z)/(p1**2 + p3**2)},
  {-((p1*p3*(p1**2 + p2**2 + p3**2))/(p1**2 + p3**2)), 0,
   -((p3**2*(p1**2 + p2**2 + p3**2))/(p1**2 + p3**2)),
   -((p1*p2*p3*z)/(p1**2 + p3**2)),p3*z,-((p2*p3**2*z)/(p1**2 + p3**2)),
   -((p3**2*(p1**2 + p2**2 + p3**2)*z)/((p1**2 + p3**2)*(s + z))), 0,
   (p1*p3*(p1**2 + p2**2 + p3**2)*z)/((p1**2 + p3**2)*(s + z))},
  {0, 0, 0, 0, 0, 0, 0, 0, 0}, 
  {(p1**2*(p1**2 + p2**2 + p3**2))/(p1**2 + p3**2), 0,
   (p1*p3*(p1**2 + p2**2 + p3**2))/(p1**2 + p3**2),
   (p1**2*p2*z)/(p1**2 + p3**2), -(p1*z), (p1*p2*p3*z)/(p1**2 + p3**2),
   (p1*p3*(p1**2 + p2**2 + p3**2)*z)/((p1**2 + p3**2)*(s + z)), 0,
   -((p1**2*(p1**2 + p2**2 + p3**2)*z)/((p1**2 + p3**2)*(s + z)))}};


           2            2     2     2     2
         P1 *( - S*Z - Z  + P1  + P2  + P3 )
C1 := {{-------------------------------------,
                        2     2
                      P1  + P3

        0,

                          2     2     2     2
         P1*P3*( - S*Z - Z  + P1  + P2  + P3 )
        ---------------------------------------,
                         2     2
                       P1  + P3

               2
          - (P1 *P2)*(S + Z)
        ---------------------,
                2     2
              P1  + P3

        P1*(S + Z),

          - (P1*P2*P3)*(S + Z)
        -----------------------,
                 2     2
               P1  + P3

                       2     2     2
          - (P1*P3)*(P1  + P2  + P3 )
        ------------------------------,
                    2     2
                  P1  + P3

        0,

           2    2     2     2
         P1 *(P1  + P2  + P3 )
        -----------------------},
                 2     2
               P1  + P3

       {0,0,0,0,0,0,0,0,0},

                          2     2     2     2
         P1*P3*( - S*Z - Z  + P1  + P2  + P3 )
       {---------------------------------------,
                         2     2
                       P1  + P3

        0,

           2            2     2     2     2
         P3 *( - S*Z - Z  + P1  + P2  + P3 )
        -------------------------------------,
                        2     2
                      P1  + P3

          - (P1*P2*P3)*(S + Z)
        -----------------------,
                 2     2
               P1  + P3

        P3*(S + Z),

                  2
          - (P2*P3 )*(S + Z)
        ---------------------,
                2     2
              P1  + P3

              2    2     2     2
          - P3 *(P1  + P2  + P3 )
        --------------------------,
                  2     2
                P1  + P3

        0,

                  2     2     2
         P1*P3*(P1  + P2  + P3 )
        -------------------------},
                  2     2
                P1  + P3

               2
          - (P1 *P2)*(S + Z)
       {---------------------,
                2     2
              P1  + P3

        0,

          - (P1*P2*P3)*(S + Z)
        -----------------------,
                 2     2
               P1  + P3

           2   2
         P1 *P2 *( - S - 2*Z)
        ----------------------,
                 2     2
            Z*(P1  + P3 )

         P1*P2*(S + 2*Z)
        -----------------,
                Z

              2
         P1*P2 *P3*( - S - 2*Z)
        ------------------------,
                  2     2
             Z*(P1  + P3 )

          - Z*P1*P2*P3
        ---------------,
             2     2
           P1  + P3

        0,

             2
         Z*P1 *P2
        -----------},
           2     2
         P1  + P3

       {P1*(S + Z),

        0,

        P3*(S + Z),

         P1*P2*(S + 2*Z)
        -----------------,
                Z

                2       2         2         2
          - S*P1  - S*P3  - 2*Z*P1  - 2*Z*P3
        --------------------------------------,
                          Z

         P2*P3*(S + 2*Z)
        -----------------,
                Z

        Z*P3,

        0,

         - Z*P1},

          - (P1*P2*P3)*(S + Z)
       {-----------------------,
                 2     2
               P1  + P3

        0,

                  2
          - (P2*P3 )*(S + Z)
        ---------------------,
                2     2
              P1  + P3

              2
         P1*P2 *P3*( - S - 2*Z)
        ------------------------,
                  2     2
             Z*(P1  + P3 )

         P2*P3*(S + 2*Z)
        -----------------,
                Z

           2   2
         P2 *P3 *( - S - 2*Z)
        ----------------------,
                 2     2
            Z*(P1  + P3 )

                   2
          - Z*P2*P3
        -------------,
            2     2
          P1  + P3

        0,

         Z*P1*P2*P3
        ------------},
           2     2
         P1  + P3

                       2     2     2
          - (P1*P3)*(P1  + P2  + P3 )
       {------------------------------,
                    2     2
                  P1  + P3

        0,

              2    2     2     2
          - P3 *(P1  + P2  + P3 )
        --------------------------,
                  2     2
                P1  + P3

          - Z*P1*P2*P3
        ---------------,
             2     2
           P1  + P3

        Z*P3,

                   2
          - Z*P2*P3
        -------------,
            2     2
          P1  + P3

                 2     2     2     2
          - (Z*P3 )*(P1  + P2  + P3 )
        -------------------------------,
             2       2       2       2
         S*P1  + S*P3  + Z*P1  + Z*P3

        0,

                      2     2     2
           Z*P1*P3*(P1  + P2  + P3 )
        -------------------------------},
             2       2       2       2
         S*P1  + S*P3  + Z*P1  + Z*P3

       {0,0,0,0,0,0,0,0,0},

           2    2     2     2
         P1 *(P1  + P2  + P3 )
       {-----------------------,
                 2     2
               P1  + P3

        0,

                  2     2     2
         P1*P3*(P1  + P2  + P3 )
        -------------------------,
                  2     2
                P1  + P3

             2
         Z*P1 *P2
        -----------,
           2     2
         P1  + P3

         - Z*P1,

         Z*P1*P2*P3
        ------------,
           2     2
         P1  + P3

                      2     2     2
           Z*P1*P3*(P1  + P2  + P3 )
        -------------------------------,
             2       2       2       2
         S*P1  + S*P3  + Z*P1  + Z*P3

        0,

                 2     2     2     2
          - (Z*P1 )*(P1  + P2  + P3 )
        -------------------------------}}
             2       2       2       2
         S*P1  + S*P3  + Z*P1  + Z*P3


nullspace c1;


        - P1
{{1,0,-------,0,0,0,0,0,0},
        P3

 {0,1,0,0,0,0,0,0,0},

              - P1
 {0,0,0,1,0,-------,0,0,0},
              P3

               2     2
             P1  + P3
 {0,0,0,0,1,-----------,0,0,0},
               P2*P3

                   P3
 {0,0,0,0,0,0,1,0,----},
                   P1

 {0,0,0,0,0,0,0,1,0}}

 
d1 := mat
(((p1**2*(p1**2 + p2**2 + p3**2 - s*z - z**2))/(p1**2 + p3**2), 0,
   (p1*p3*(p1**2 + p2**2 + p3**2 - s*z - z**2))/(p1**2 + p3**2),
   -((p1**2*p2*(s + z))/(p1**2 + p3**2)), p1*(s + z),
   -((p1*p2*p3*(s + z))/(p1**2 + p3**2)),
   -((p1*p3*(p1**2 + p2**2 + p3**2))/(p1**2 + p3**2)), 0,
   (p1**2*(p1**2 + p2**2 + p3**2))/(p1**2 + p3**2)),
   (0, 0, 0, 0, 0, 0, 0, 0, 0),
  ((p1*p3*(p1**2 + p2**2 + p3**2 - s*z - z**2))/(p1**2 + p3**2), 0,
   (p3**2*(p1**2 + p2**2 + p3**2 - s*z - z**2))/(p1**2 + p3**2),
   -((p1*p2*p3*(s + z))/(p1**2 + p3**2)), p3*(s + z),
   -((p2*p3**2*(s + z))/(p1**2 + p3**2)),
   -((p3**2*(p1**2 + p2**2 + p3**2))/(p1**2 + p3**2)), 0,
   (p1*p3*(p1**2 + p2**2 + p3**2))/(p1**2 + p3**2)),
  ( ((p1**2*p2*(s + z))/(p1**2 + p3**2)), 0,
   -((p1*p2*p3*(s + z))/(p1**2 + p3**2)),
   -((p1**2*p2**2*(s + 2*z))/((p1**2 + p3**2)*z)), (p1*p2*(s + 2*z))/z,
   -((p1*p2**2*p3*(s + 2*z))/((p1**2 + p3**2)*z)),
   -((p1*p2*p3*z)/(p1**2 + p3**2)), 0, (p1**2*p2*z)/(p1**2 + p3**2)),
  (p1*(s + z), 0, p3*(s + z), (p1*p2*(s + 2*z))/z,
   -(((p1**2 + p3**2)*(s + 2*z))/z),(p2*p3*(s + 2*z))/z,p3*z,0,-(p1*z)),
  (-((p1*p2*p3*(s + z))/(p1**2 + p3**2)), 0,
   -((p2*p3**2*(s + z))/(p1**2 + p3**2)),
   -((p1*p2**2*p3*(s + 2*z))/((p1**2 + p3**2)*z)), (p2*p3*(s + 2*z))/z,
   -((p2**2*p3**2*(s + 2*z))/((p1**2 + p3**2)*z)),
   -((p2*p3**2*z)/(p1**2 + p3**2)), 0, (p1*p2*p3*z)/(p1**2 + p3**2)),
  (-((p1*p3*(p1**2 + p2**2 + p3**2))/(p1**2 + p3**2)), 0,
   -((p3**2*(p1**2 + p2**2 + p3**2))/(p1**2 + p3**2)),
   -((p1*p2*p3*z)/(p1**2 + p3**2)),p3*z,-((p2*p3**2*z)/(p1**2 + p3**2)),
   -((p3**2*(p1**2 + p2**2 + p3**2)*z)/((p1**2 + p3**2)*(s + z))), 0,
   (p1*p3*(p1**2 + p2**2 + p3**2)*z)/((p1**2 + p3**2)*(s + z))),
  (0, 0, 0, 0, 0, 0, 0, 0, 0), 
   ((p1**2*(p1**2 + p2**2 + p3**2))/(p1**2 + p3**2), 0,
   (p1*p3*(p1**2 + p2**2 + p3**2))/(p1**2 + p3**2),
   (p1**2*p2*z)/(p1**2 + p3**2), -(p1*z), (p1*p2*p3*z)/(p1**2 + p3**2),
   (p1*p3*(p1**2 + p2**2 + p3**2)*z)/((p1**2 + p3**2)*(s + z)), 0,
   -((p1**2*(p1**2 + p2**2 + p3**2)*z)/((p1**2 + p3**2)*(s + z)))));


              2            2     2     2     2
            P1 *( - S*Z - Z  + P1  + P2  + P3 )
D1 := MAT((-------------------------------------,0,
                           2     2
                         P1  + P3

                             2     2     2     2
            P1*P3*( - S*Z - Z  + P1  + P2  + P3 )
           ---------------------------------------,
                            2     2
                          P1  + P3

                  2
             - (P1 *P2)*(S + Z)               - (P1*P2*P3)*(S + Z)
           ---------------------,P1*(S + Z),-----------------------,
                   2     2                           2     2
                 P1  + P3                          P1  + P3

                          2     2     2        2    2     2     2
             - (P1*P3)*(P1  + P2  + P3 )     P1 *(P1  + P2  + P3 )
           ------------------------------,0,-----------------------),
                       2     2                       2     2
                     P1  + P3                      P1  + P3

          (0,0,0,0,0,0,0,0,0),

                             2     2     2     2
            P1*P3*( - S*Z - Z  + P1  + P2  + P3 )
          (---------------------------------------,0,
                            2     2
                          P1  + P3

              2            2     2     2     2
            P3 *( - S*Z - Z  + P1  + P2  + P3 )
           -------------------------------------,
                           2     2
                         P1  + P3

                                                        2
             - (P1*P2*P3)*(S + Z)               - (P2*P3 )*(S + Z)
           -----------------------,P3*(S + Z),---------------------,
                    2     2                           2     2
                  P1  + P3                          P1  + P3

                 2    2     2     2               2     2     2
             - P3 *(P1  + P2  + P3 )     P1*P3*(P1  + P2  + P3 )
           --------------------------,0,-------------------------),
                     2     2                      2     2
                   P1  + P3                     P1  + P3

              2
            P1 *P2*(S + Z)      - (P1*P2*P3)*(S + Z)
          (----------------,0,-----------------------,
                2     2                2     2
              P1  + P3               P1  + P3

              2   2
            P1 *P2 *( - S - 2*Z)   P1*P2*(S + 2*Z)
           ----------------------,-----------------,
                    2     2               Z
               Z*(P1  + P3 )

                 2                                         2
            P1*P2 *P3*( - S - 2*Z)    - Z*P1*P2*P3     Z*P1 *P2
           ------------------------,---------------,0,-----------),
                     2     2             2     2         2     2
                Z*(P1  + P3 )          P1  + P3        P1  + P3

                                    P1*P2*(S + 2*Z)
          (P1*(S + Z),0,P3*(S + Z),-----------------,
                                           Z

                   2       2         2         2
             - S*P1  - S*P3  - 2*Z*P1  - 2*Z*P3    P2*P3*(S + 2*Z)
           --------------------------------------,-----------------,
                             Z                            Z

           Z*P3,0, - Z*P1),

                                               2
             - (P1*P2*P3)*(S + Z)      - (P2*P3 )*(S + Z)
          (-----------------------,0,---------------------,
                    2     2                  2     2
                  P1  + P3                 P1  + P3

                 2
            P1*P2 *P3*( - S - 2*Z)   P2*P3*(S + 2*Z)
           ------------------------,-----------------,
                     2     2                Z
                Z*(P1  + P3 )

              2   2                          2
            P2 *P3 *( - S - 2*Z)    - Z*P2*P3      Z*P1*P2*P3
           ----------------------,-------------,0,------------),
                    2     2           2     2        2     2
               Z*(P1  + P3 )        P1  + P3       P1  + P3

                          2     2     2
             - (P1*P3)*(P1  + P2  + P3 )
          (------------------------------,0,
                       2     2
                     P1  + P3

                 2    2     2     2
             - P3 *(P1  + P2  + P3 )    - Z*P1*P2*P3
           --------------------------,---------------,Z*P3,
                     2     2               2     2
                   P1  + P3              P1  + P3

                      2           2     2     2     2
             - Z*P2*P3     - (Z*P3 )*(P1  + P2  + P3 )
           -------------,-------------------------------,0,
               2     2        2       2       2       2
             P1  + P3     S*P1  + S*P3  + Z*P1  + Z*P3

                         2     2     2
              Z*P1*P3*(P1  + P2  + P3 )
           -------------------------------),
                2       2       2       2
            S*P1  + S*P3  + Z*P1  + Z*P3

          (0,0,0,0,0,0,0,0,0),

              2    2     2     2               2     2     2
            P1 *(P1  + P2  + P3 )     P1*P3*(P1  + P2  + P3 )
          (-----------------------,0,-------------------------,
                    2     2                    2     2
                  P1  + P3                   P1  + P3

                2
            Z*P1 *P2            Z*P1*P2*P3
           -----------, - Z*P1,------------,
              2     2             2     2
            P1  + P3            P1  + P3

                         2     2     2
              Z*P1*P3*(P1  + P2  + P3 )
           -------------------------------,0,
                2       2       2       2
            S*P1  + S*P3  + Z*P1  + Z*P3

                    2     2     2     2
             - (Z*P1 )*(P1  + P2  + P3 )
           -------------------------------))
                2       2       2       2
            S*P1  + S*P3  + Z*P1  + Z*P3



nullspace d1;


{

 [0]
 [ ]
 [1]
 [ ]
 [0]
 [ ]
 [0]
 [ ]
 [0]
 [ ]
 [0]
 [ ]
 [0]
 [ ]
 [0]
 [ ]
 [0]

 ,


 [   0   ]
 [       ]
 [   0   ]
 [       ]
 [   0   ]
 [       ]
 [   1   ]
 [       ]
 [   0   ]
 [       ]
 [  - P1 ]
 [-------]
 [  P3   ]
 [       ]
 [   0   ]
 [       ]
 [   0   ]
 [       ]
 [   0   ]

 ,


 [     0     ]
 [           ]
 [     0     ]
 [           ]
 [     0     ]
 [           ]
 [     0     ]
 [           ]
 [     1     ]
 [           ]
 [   2     2 ]
 [ P1  + P3  ]
 [-----------]
 [   P2*P3   ]
 [           ]
 [     0     ]
 [           ]
 [     0     ]
 [           ]
 [     0     ]

 ,


 [ 0  ]
 [    ]
 [ 0  ]
 [    ]
 [ 0  ]
 [    ]
 [ 0  ]
 [    ]
 [ 0  ]
 [    ]
 [ 0  ]
 [    ]
 [ 1  ]
 [    ]
 [ 0  ]
 [    ]
 [ P3 ]
 [----]
 [ P1 ]

 ,


 [0]
 [ ]
 [0]
 [ ]
 [0]
 [ ]
 [0]
 [ ]
 [0]
 [ ]
 [0]
 [ ]
 [0]
 [ ]
 [1]
 [ ]
 [0]

 }



% The following example, by Kenton Yee, was discussed extensively by
% the sci.math.symbolic newsgroup.

m := mat((e^(-1), e^(-1), e^(-1), e^(-1), e^(-1), e^(-1), e^(-1), 0),
         (1, 1, 1, 1, 1, 1, 0, 1),(1, 1, 1, 1, 1, 0, 1, 1),
         (1, 1, 1, 1, 0, 1, 1, 1),(1, 1, 1, 0, 1, 1, 1, 1),
         (1, 1, 0, 1, 1, 1, 1, 1),(1, 0, 1, 1, 1, 1, 1, 1),
         (0, e, e, e, e, e, e, e));


     [ 1    1    1    1    1    1    1    ]
     [---  ---  ---  ---  ---  ---  ---  0]
     [ E    E    E    E    E    E    E    ]
     [                                    ]
     [ 1    1    1    1    1    1    0   1]
     [                                    ]
     [ 1    1    1    1    1    0    1   1]
     [                                    ]
M := [ 1    1    1    1    0    1    1   1]
     [                                    ]
     [ 1    1    1    0    1    1    1   1]
     [                                    ]
     [ 1    1    0    1    1    1    1   1]
     [                                    ]
     [ 1    0    1    1    1    1    1   1]
     [                                    ]
     [ 0    E    E    E    E    E    E   E]



eig := mateigen(m,x);


EIG := {{X - 1,

         3,


         [        0        ]
         [                 ]
         [ - ARBCOMPLEX(20)]
         [                 ]
         [ - ARBCOMPLEX(19)]
         [                 ]
         [ - ARBCOMPLEX(18)]
         [                 ]
         [ ARBCOMPLEX(18)  ]
         [                 ]
         [ ARBCOMPLEX(19)  ]
         [                 ]
         [ ARBCOMPLEX(20)  ]
         [                 ]
         [        0        ]

         },

        {X + 1,

         3,


               ARBCOMPLEX(23)
         MAT((----------------),
                     E

             (ARBCOMPLEX(22)),

             (ARBCOMPLEX(21)),

             (( - ARBCOMPLEX(23)*E - ARBCOMPLEX(23)

                - 2*ARBCOMPLEX(22)*E - 2*ARBCOMPLEX(21)*E)/(2*E)),

             (( - ARBCOMPLEX(23)*E - ARBCOMPLEX(23)

                - 2*ARBCOMPLEX(22)*E - 2*ARBCOMPLEX(21)*E)/(2*E)),

             (ARBCOMPLEX(21)),

             (ARBCOMPLEX(22)),

             (ARBCOMPLEX(23)))

},

             2        2
        { - E *X + E*X  - 6*E*X + 7*E - X,

         1,


                                  8         7        7        6
         MAT(((6*ARBCOMPLEX(24)*(E *X + 23*E *X - 7*E  + 179*E *X

                          6        5          5        4          4
                   - 119*E  + 565*E *X - 581*E  + 768*E *X - 890*E

                          3          3        2          2
                   + 565*E *X - 581*E  + 179*E *X - 119*E  + 23*E*X

                                 3   8         7        7        6
                   - 7*E + X))/(E *(E *X + 30*E *X - 7*E  + 333*E *X

                            6         5           5         4
                     - 168*E  + 1692*E *X - 1365*E  + 4023*E *X

                             4         3           3         2
                     - 4368*E  + 4470*E *X - 5145*E  + 2663*E *X

                             2
                     - 2520*E  + 576*E*X - 251*E + 36*X))),

                                9         8        8        7
             ((ARBCOMPLEX(24)*(E *X + 29*E *X - 7*E  + 310*E *X

                          7         6           6         5
                   - 161*E  + 1520*E *X - 1246*E  + 3577*E *X

                           5         4           4         3
                   - 3836*E  + 4283*E *X - 4795*E  + 2988*E *X

                           3        2          2
                   - 3065*E  + 978*E *X - 672*E  + 132*E*X - 42*E

                             2   8         7        7        6
                   + 6*X))/(E *(E *X + 30*E *X - 7*E  + 333*E *X

                            6         5           5         4
                     - 168*E  + 1692*E *X - 1365*E  + 4023*E *X

                             4         3           3         2
                     - 4368*E  + 4470*E *X - 5145*E  + 2663*E *X

                             2
                     - 2520*E  + 576*E*X - 251*E + 36*X))),

                                9         8        8        7
             ((ARBCOMPLEX(24)*(E *X + 29*E *X - 7*E  + 310*E *X

                          7         6           6         5
                   - 161*E  + 1520*E *X - 1246*E  + 3577*E *X

                           5         4           4         3
                   - 3836*E  + 4283*E *X - 4795*E  + 2988*E *X

                           3        2          2
                   - 3065*E  + 978*E *X - 672*E  + 132*E*X - 42*E

                             2   8         7        7        6
                   + 6*X))/(E *(E *X + 30*E *X - 7*E  + 333*E *X

                            6         5           5         4
                     - 168*E  + 1692*E *X - 1365*E  + 4023*E *X

                             4         3           3         2
                     - 4368*E  + 4470*E *X - 5145*E  + 2663*E *X

                             2
                     - 2520*E  + 576*E*X - 251*E + 36*X))),

                                9         8        8        7
             ((ARBCOMPLEX(24)*(E *X + 29*E *X - 7*E  + 310*E *X

                          7         6           6         5
                   - 161*E  + 1520*E *X - 1246*E  + 3577*E *X

                           5         4           4         3
                   - 3836*E  + 4283*E *X - 4795*E  + 2988*E *X

                           3        2          2
                   - 3065*E  + 978*E *X - 672*E  + 132*E*X - 42*E

                             2   8         7        7        6
                   + 6*X))/(E *(E *X + 30*E *X - 7*E  + 333*E *X

                            6         5           5         4
                     - 168*E  + 1692*E *X - 1365*E  + 4023*E *X

                             4         3           3         2
                     - 4368*E  + 4470*E *X - 5145*E  + 2663*E *X

                             2
                     - 2520*E  + 576*E*X - 251*E + 36*X))),

                                9         8        8        7
             ((ARBCOMPLEX(24)*(E *X + 29*E *X - 7*E  + 310*E *X

                          7         6           6         5
                   - 161*E  + 1520*E *X - 1246*E  + 3577*E *X

                           5         4           4         3
                   - 3836*E  + 4283*E *X - 4795*E  + 2988*E *X

                           3        2          2
                   - 3065*E  + 978*E *X - 672*E  + 132*E*X - 42*E

                             2   8         7        7        6
                   + 6*X))/(E *(E *X + 30*E *X - 7*E  + 333*E *X

                            6         5           5         4
                     - 168*E  + 1692*E *X - 1365*E  + 4023*E *X

                             4         3           3         2
                     - 4368*E  + 4470*E *X - 5145*E  + 2663*E *X

                             2
                     - 2520*E  + 576*E*X - 251*E + 36*X))),

                                9         8        8        7
             ((ARBCOMPLEX(24)*(E *X + 29*E *X - 7*E  + 310*E *X

                          7         6           6         5
                   - 161*E  + 1520*E *X - 1246*E  + 3577*E *X

                           5         4           4         3
                   - 3836*E  + 4283*E *X - 4795*E  + 2988*E *X

                           3        2          2
                   - 3065*E  + 978*E *X - 672*E  + 132*E*X - 42*E

                             2   8         7        7        6
                   + 6*X))/(E *(E *X + 30*E *X - 7*E  + 333*E *X

                            6         5           5         4
                     - 168*E  + 1692*E *X - 1365*E  + 4023*E *X

                             4         3           3         2
                     - 4368*E  + 4470*E *X - 5145*E  + 2663*E *X

                             2
                     - 2520*E  + 576*E*X - 251*E + 36*X))),

                                9         8        8        7
             ((ARBCOMPLEX(24)*(E *X + 29*E *X - 7*E  + 310*E *X

                          7         6           6         5
                   - 161*E  + 1520*E *X - 1246*E  + 3577*E *X

                           5         4           4         3
                   - 3836*E  + 4283*E *X - 4795*E  + 2988*E *X

                           3        2          2
                   - 3065*E  + 978*E *X - 672*E  + 132*E*X - 42*E

                             2   8         7        7        6
                   + 6*X))/(E *(E *X + 30*E *X - 7*E  + 333*E *X

                            6         5           5         4
                     - 168*E  + 1692*E *X - 1365*E  + 4023*E *X

                             4         3           3         2
                     - 4368*E  + 4470*E *X - 5145*E  + 2663*E *X

                             2
                     - 2520*E  + 576*E*X - 251*E + 36*X))),

             (ARBCOMPLEX(24)))

}}


% Now check the eigenvectors and calculate the eigenvalues in the
% respective eigenspaces:

factor expt;



for each eispace in eig do
  begin scalar eivaleq,eival,eivec;
    eival := solve(first eispace,x);
    for each soln in eival do
      <<eival := rhs soln;
        eivec := third eispace;
        eivec := sub(soln,eivec);
        write "eigenvalue = ", eival;
        write "check of eigen equation: ", 
              m*eivec - eival*eivec>>
  end;


eigenvalue = 1

check of eigen equation: 

[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]


eigenvalue = -1

check of eigen equation: 

[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]


                    4       3       2                2
              SQRT(E  + 12*E  + 10*E  + 12*E + 1) + E  + 6*E + 1
eigenvalue = ----------------------------------------------------
                                     2*E

check of eigen equation: 

[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]


                       4       3       2                2
               - SQRT(E  + 12*E  + 10*E  + 12*E + 1) + E  + 6*E + 1
eigenvalue = -------------------------------------------------------
                                       2*E

check of eigen equation: 

[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]



% For the special choice:

let e = -7 + sqrt 48;



% we get only 7 eigenvectors.

eig := mateigen(m,x);


EIG := {{X + 1,

         4,


               ARBCOMPLEX(27)
         MAT((----------------),
               4*SQRT(3) - 7

             (ARBCOMPLEX(26)),

             (ARBCOMPLEX(25)),

             ((2*SQRT(3)*( - ARBCOMPLEX(27) - 2*ARBCOMPLEX(26)

                   - 2*ARBCOMPLEX(25)) + 3*ARBCOMPLEX(27)

                + 7*ARBCOMPLEX(26) + 7*ARBCOMPLEX(25))/(4*SQRT(3) - 7

                 )),

             ((2*SQRT(3)*( - ARBCOMPLEX(27) - 2*ARBCOMPLEX(26)

                   - 2*ARBCOMPLEX(25)) + 3*ARBCOMPLEX(27)

                + 7*ARBCOMPLEX(26) + 7*ARBCOMPLEX(25))/(4*SQRT(3) - 7

                 )),

             (ARBCOMPLEX(25)),

             (ARBCOMPLEX(26)),

             (ARBCOMPLEX(27)))

},

        {X - 1,

         3,


         [        0        ]
         [                 ]
         [ - ARBCOMPLEX(30)]
         [                 ]
         [ - ARBCOMPLEX(29)]
         [                 ]
         [ - ARBCOMPLEX(28)]
         [                 ]
         [ ARBCOMPLEX(28)  ]
         [                 ]
         [ ARBCOMPLEX(29)  ]
         [                 ]
         [ ARBCOMPLEX(30)  ]
         [                 ]
         [        0        ]

         },

        {X + 7,

         1,


         [                 ARBCOMPLEX(31)                   ]
         [                -----------------                 ]
         [                 56*SQRT(3) - 97                  ]
         [                                                  ]
         [  - 14*SQRT(3)*ARBCOMPLEX(31) + 24*ARBCOMPLEX(31) ]
         [--------------------------------------------------]
         [                168*SQRT(3) - 291                 ]
         [                                                  ]
         [  - 14*SQRT(3)*ARBCOMPLEX(31) + 24*ARBCOMPLEX(31) ]
         [--------------------------------------------------]
         [                168*SQRT(3) - 291                 ]
         [                                                  ]
         [  - 14*SQRT(3)*ARBCOMPLEX(31) + 24*ARBCOMPLEX(31) ]
         [--------------------------------------------------]
         [                168*SQRT(3) - 291                 ]
         [                                                  ]
         [  - 14*SQRT(3)*ARBCOMPLEX(31) + 24*ARBCOMPLEX(31) ]
         [--------------------------------------------------]
         [                168*SQRT(3) - 291                 ]
         [                                                  ]
         [  - 14*SQRT(3)*ARBCOMPLEX(31) + 24*ARBCOMPLEX(31) ]
         [--------------------------------------------------]
         [                168*SQRT(3) - 291                 ]
         [                                                  ]
         [  - 14*SQRT(3)*ARBCOMPLEX(31) + 24*ARBCOMPLEX(31) ]
         [--------------------------------------------------]
         [                168*SQRT(3) - 291                 ]
         [                                                  ]
         [                  ARBCOMPLEX(31)                  ]

         }}


for each eispace in eig do
  begin scalar eivaleq,eival,eivec;
    eival := solve(first eispace,x);
    for each soln in eival do
      <<eival := rhs soln;
        eivec := third eispace;
        eivec := sub(soln,eivec);
        write "eigenvalue = ", eival;
        write "check of eigen equation: ", 
              m*eivec - eival*eivec>>
  end;


eigenvalue = -1

check of eigen equation: 

[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]


eigenvalue = 1

check of eigen equation: 

[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]


eigenvalue = -7

check of eigen equation: 

[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]



% The same behaviour for this choice of e.

clear e;

 let e = -7 - sqrt 48;



% we get only 7 eigenvectors.

eig := mateigen(m,x);


EIG := {{X + 1,

         4,


                - ARBCOMPLEX(34)
         MAT((-------------------),
                 4*SQRT(3) + 7

             (ARBCOMPLEX(33)),

             (ARBCOMPLEX(32)),

             ((2*SQRT(3)*( - ARBCOMPLEX(34) - 2*ARBCOMPLEX(33)

                   - 2*ARBCOMPLEX(32)) - 3*ARBCOMPLEX(34)

                - 7*ARBCOMPLEX(33) - 7*ARBCOMPLEX(32))/(4*SQRT(3) + 7

                 )),

             ((2*SQRT(3)*( - ARBCOMPLEX(34) - 2*ARBCOMPLEX(33)

                   - 2*ARBCOMPLEX(32)) - 3*ARBCOMPLEX(34)

                - 7*ARBCOMPLEX(33) - 7*ARBCOMPLEX(32))/(4*SQRT(3) + 7

                 )),

             (ARBCOMPLEX(32)),

             (ARBCOMPLEX(33)),

             (ARBCOMPLEX(34)))

},

        {X - 1,

         3,


         [        0        ]
         [                 ]
         [ - ARBCOMPLEX(37)]
         [                 ]
         [ - ARBCOMPLEX(36)]
         [                 ]
         [ - ARBCOMPLEX(35)]
         [                 ]
         [ ARBCOMPLEX(35)  ]
         [                 ]
         [ ARBCOMPLEX(36)  ]
         [                 ]
         [ ARBCOMPLEX(37)  ]
         [                 ]
         [        0        ]

         },

        {X + 7,

         1,


         [                 - ARBCOMPLEX(38)                 ]
         [               -------------------                ]
         [                 56*SQRT(3) + 97                  ]
         [                                                  ]
         [  - 14*SQRT(3)*ARBCOMPLEX(38) - 24*ARBCOMPLEX(38) ]
         [--------------------------------------------------]
         [                168*SQRT(3) + 291                 ]
         [                                                  ]
         [  - 14*SQRT(3)*ARBCOMPLEX(38) - 24*ARBCOMPLEX(38) ]
         [--------------------------------------------------]
         [                168*SQRT(3) + 291                 ]
         [                                                  ]
         [  - 14*SQRT(3)*ARBCOMPLEX(38) - 24*ARBCOMPLEX(38) ]
         [--------------------------------------------------]
         [                168*SQRT(3) + 291                 ]
         [                                                  ]
         [  - 14*SQRT(3)*ARBCOMPLEX(38) - 24*ARBCOMPLEX(38) ]
         [--------------------------------------------------]
         [                168*SQRT(3) + 291                 ]
         [                                                  ]
         [  - 14*SQRT(3)*ARBCOMPLEX(38) - 24*ARBCOMPLEX(38) ]
         [--------------------------------------------------]
         [                168*SQRT(3) + 291                 ]
         [                                                  ]
         [  - 14*SQRT(3)*ARBCOMPLEX(38) - 24*ARBCOMPLEX(38) ]
         [--------------------------------------------------]
         [                168*SQRT(3) + 291                 ]
         [                                                  ]
         [                  ARBCOMPLEX(38)                  ]

         }}


for each eispace in eig do
  begin scalar eivaleq,eival,eivec;
    eival := solve(first eispace,x);
    for each soln in eival do
      <<eival := rhs soln;
        eivec := third eispace;
        eivec := sub(soln,eivec);
        write "eigenvalue = ", eival;
        write "check of eigen equation: ", 
              m*eivec - eival*eivec>>
  end;


eigenvalue = -1

check of eigen equation: 

[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]


eigenvalue = 1

check of eigen equation: 

[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]


eigenvalue = -7

check of eigen equation: 

[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]




% For this choice of values

clear e;

 let e = 1;



% the eigenvalue 1 becomes 4-fold degenerate. However, we get a complete
% span of 8 eigenvectors.

eig := mateigen(m,x);


EIG := {{X - 1,

         4,


         [ - ARBCOMPLEX(42)]
         [                 ]
         [ - ARBCOMPLEX(41)]
         [                 ]
         [ - ARBCOMPLEX(40)]
         [                 ]
         [ - ARBCOMPLEX(39)]
         [                 ]
         [ ARBCOMPLEX(39)  ]
         [                 ]
         [ ARBCOMPLEX(40)  ]
         [                 ]
         [ ARBCOMPLEX(41)  ]
         [                 ]
         [ ARBCOMPLEX(42)  ]

         },

        {X + 1,

         3,


         [                   ARBCOMPLEX(45)                    ]
         [                                                     ]
         [                   ARBCOMPLEX(44)                    ]
         [                                                     ]
         [                   ARBCOMPLEX(43)                    ]
         [                                                     ]
         [ - (ARBCOMPLEX(45) + ARBCOMPLEX(44) + ARBCOMPLEX(43))]
         [                                                     ]
         [ - (ARBCOMPLEX(45) + ARBCOMPLEX(44) + ARBCOMPLEX(43))]
         [                                                     ]
         [                   ARBCOMPLEX(43)                    ]
         [                                                     ]
         [                   ARBCOMPLEX(44)                    ]
         [                                                     ]
         [                   ARBCOMPLEX(45)                    ]

         },

        {X - 7,

         1,


         [ARBCOMPLEX(46)]
         [              ]
         [ARBCOMPLEX(46)]
         [              ]
         [ARBCOMPLEX(46)]
         [              ]
         [ARBCOMPLEX(46)]
         [              ]
         [ARBCOMPLEX(46)]
         [              ]
         [ARBCOMPLEX(46)]
         [              ]
         [ARBCOMPLEX(46)]
         [              ]
         [ARBCOMPLEX(46)]

         }}


for each eispace in eig do
  begin scalar eivaleq,eival,eivec;
    eival := solve(first eispace,x);
    for each soln in eival do
      <<eival := rhs soln;
        eivec := third eispace;
        eivec := sub(soln,eivec);
        write "eigenvalue = ", eival;
        write "check of eigen equation: ", 
              m*eivec - eival*eivec>>
  end;


eigenvalue = 1

check of eigen equation: 

[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]


eigenvalue = -1

check of eigen equation: 

[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]


eigenvalue = 7

check of eigen equation: 

[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]



ma := mat((1,a),(0,b));


      [1  A]
MA := [    ]
      [0  B]



% case 1:

let a = 0;



mateigen(ma,x);


{{X - 1,1,

  [ARBCOMPLEX(47)]
  [              ]
  [      0       ]

  },

 { - B + X,1,

  [      0       ]
  [              ]
  [ARBCOMPLEX(48)]

  }}


% case 2:

clear a;

 let a = 0, b = 1;



mateigen(ma,x);


{{X - 1,2,

  [ARBCOMPLEX(49)]
  [              ]
  [ARBCOMPLEX(50)]

  }}


% case 3:

clear a,b;

 

mateigen(ma,x);


{{ - B + X,

  1,


  [ ARBCOMPLEX(51)*A ]
  [------------------]
  [      B - 1       ]
  [                  ]
  [  ARBCOMPLEX(51)  ]

  },

 {X - 1,1,

  [ARBCOMPLEX(52)]
  [              ]
  [      0       ]

  }}


% case 4:

let b = 1;

 

mateigen(ma,x);


{{X - 1,2,

  [ARBCOMPLEX(53)]
  [              ]
  [      0       ]

  }}


end;

4: 4: 
Quitting
Sat Jun 29 13:45:44 PDT 1991