O15-C911
> restart:with(linalg):
Warning, new definition for norm
Warning, new definition for trace
> m:=matrix(3,3,[-1,a,a,1,-1,0,-1,0,-1]);

                             [-1    a     a ]
                             [              ]
                        m := [ 1    -1    0 ]
                             [              ]
                             [-1    0     -1]

> eigenvects(m);

                        [-1, 3, {[0, -1, 1]}]

> jordan(m,'p');

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

> evalm(p);

                            [0     a    0]
                            [            ]
                            [a     0    0]
                            [            ]
                            [-a    0    1]

> m:=proc(n)
> local k,res;
> res:=matrix(n,n,0);
> for k from 2 to n do res[k,k]:=-1;res[k,1]:=cos(k*Pi);res[1,k]:=a od;
> res[1,1]:=-1;
> res;end;

  m := proc(n)
local k, res;
    res := matrix(n, n, 0);
    for k from 2 to n do
        res[k, k] := -1;
        res[k, 1] := cos(k*Pi);
        res[1, k] := a
    od;
    res[1, 1] := -1;
    res
end

> for k from 4 to 7 do eigenvals(m(k)) od;

                  -1 + sqrt(a), -1 - sqrt(a), -1, -1


                          -1, -1, -1, -1, -1


              -1 + sqrt(a), -1 - sqrt(a), -1, -1, -1, -1


                      -1, -1, -1, -1, -1, -1, -1

> for k from 2 to 5 do eigenvects(m(2*k)) od;

  [-1 + sqrt(a), 1, {[sqrt(a), 1, -1, 1]}],

        [-1 - sqrt(a), 1, {[-sqrt(a), 1, -1, 1]}],

        [-1, 2, {[0, -1, 1, 0], [0, -1, 0, 1]}]


  [-1, 4, {[0, -1, 0, 1, 0, 0], [0, -1, 0, 0, 1, 0],

        [0, -1, 0, 0, 0, 1], [0, -1, 1, 0, 0, 0]}],

        [-1 + sqrt(a), 1, {[sqrt(a), 1, -1, 1, -1, 1]}],

        [-1 - sqrt(a), 1, {[-sqrt(a), 1, -1, 1, -1, 1]}]


  [-1 + sqrt(a), 1, {[sqrt(a), 1, -1, 1, -1, 1, -1, 1]}],

        [-1 - sqrt(a), 1, {[-sqrt(a), 1, -1, 1, -1, 1, -1, 1]}], [-1,

        6, {[0, -1, 0, 0, 0, 0, 1, 0], [0, -1, 0, 0, 1, 0, 0, 0],

        [0, -1, 1, 0, 0, 0, 0, 0], [0, -1, 0, 1, 0, 0, 0, 0],

        [0, -1, 0, 0, 0, 1, 0, 0], [0, -1, 0, 0, 0, 0, 0, 1]}]


  [-1 + sqrt(a), 1, {[sqrt(a), 1, -1, 1, -1, 1, -1, 1, -1, 1]}], [

        -1 - sqrt(a), 1, {[-sqrt(a), 1, -1, 1, -1, 1, -1, 1, -1, 1]}]

        , [-1, 8, {[0, -1, 0, 0, 0, 1, 0, 0, 0, 0],

        [0, -1, 0, 0, 0, 0, 0, 0, 1, 0],

        [0, -1, 0, 0, 1, 0, 0, 0, 0, 0],

        [0, -1, 0, 0, 0, 0, 0, 0, 0, 1],

        [0, -1, 1, 0, 0, 0, 0, 0, 0, 0],

        [0, -1, 0, 1, 0, 0, 0, 0, 0, 0],

        [0, -1, 0, 0, 0, 0, 0, 1, 0, 0],

        [0, -1, 0, 0, 0, 0, 1, 0, 0, 0]}]

> for k from 2 to 5 do eigenvects(m(2*k+1)) od;

   [-1, 5, {[0, -1, 0, 0, 1], [0, -1, 0, 1, 0], [0, -1, 1, 0, 0]}]


  [-1, 7, {[0, 0, 0, 0, 1, -1, 0], [0, 1, 0, 0, 0, -1, 0],

        [0, 0, 0, 1, 0, -1, 0], [0, 0, 1, 0, 0, -1, 0],

        [0, 0, 0, 0, 0, -1, 1]}]


  [-1, 9, {[0, -1, 0, 0, 0, 0, 1, 0, 0], [0, -1, 0, 0, 0, 1, 0, 0, 0],

        [0, -1, 0, 0, 0, 0, 0, 1, 0], [0, -1, 0, 0, 0, 0, 0, 0, 1],

        [0, -1, 1, 0, 0, 0, 0, 0, 0], [0, -1, 0, 0, 1, 0, 0, 0, 0],

        [0, -1, 0, 1, 0, 0, 0, 0, 0]}]


  [-1, 11, {[0, 0, 0, 0, 0, 1, 0, 0, 0, 0, -1],

        [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1],

        [0, 0, 0, 0, 0, 0, 1, 0, 0, 0, -1],

        [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, -1],

        [0, 0, 0, 0, 0, 0, 0, 1, 0, 0, -1],

        [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, -1],

        [0, 0, 1, 0, 0, 0, 0, 0, 0, 0, -1],

        [0, 0, 0, 1, 0, 0, 0, 0, 0, 0, -1],

        [0, 0, 0, 0, 1, 0, 0, 0, 0, 0, -1]}]

>