> # Harj. 4 loppuviikko
> #
> # Teht. 1
> 
> with(linalg):
Warning: new definition for   norm
Warning: new definition for   trace

--------------------------------------------------------------------------------
> A:=matrix(3,3,[k,1,0,0,k,1,0,0,m]);

                                     [ k  1  0 ]
                                     [         ]
                                A := [ 0  k  1 ]
                                     [         ]
                                     [ 0  0  m ]
--------------------------------------------------------------------------------
> eigenvals(A);

                                     k, k, m
--------------------------------------------------------------------------------
> # Paalavistajan alkiot, kuten pitaakin
> Cmat1:=charmat(A,k);

                                     [ 0  -1    0   ]
                                     [              ]
                            Cmat1 := [ 0   0    -1  ]
                                     [              ]
                                     [ 0   0  k - m ]
--------------------------------------------------------------------------------
> Cmat2:=charmat(A,m);

                                   [ m - k    -1    0 ]
                                   [                  ]
                          Cmat2 := [   0    m - k  -1 ]
                                   [                  ]
                                   [   0      0     0 ]
--------------------------------------------------------------------------------
> gaussjord(Cmat1);

                                   [ 0  1  0 ]
                                   [         ]
                                   [ 0  0  1 ]
                                   [         ]
                                   [ 0  0  0 ]
--------------------------------------------------------------------------------
> # Tasta nahdaan:
> x1[1]:=alpha;x1[2]:=0;x1[3]:=0;

                                 x1[1] := alpha

                                   x1[2] := 0

                                   x1[3] := 0
--------------------------------------------------------------------------------
> # Siis 
> lambda=k;

                                   lambda = k
> # on ominaisarvo, jolla
> k[g](lambda)<K[a](lambda);

                           k[g](lambda) < K[a](lambda)
--------------------------------------------------------------------------------
> # Tasta seuraa, etta A on defektiivinen.
> # Katsotaan joka tapauksessa m:aa vastaava om. vekt.:
> gaussjord(Cmat2);

                              [             1    ]
                              [ 1  0  - -------- ]
                              [                2 ]
                              [         (k - m)  ]
                              [                  ]
                              [            1     ]
                              [ 0  1     -----   ]
                              [          k - m   ]
                              [                  ]
                              [ 0  0       0     ]
--------------------------------------------------------------------------------
> # Tasta nahdaan:
> 
--------------------------------------------------------------------------------
> x2[3]:=alpha;x2[3]:=-1/(k-m);x2[1]:=1/(k-m)^2;

                                 x2[3] := alpha

                                             1
                                x2[3] := - -----
                                           k - m

                                             1
                                x2[1] := --------
                                                2
                                         (k - m)
--------------------------------------------------------------------------------
>