> with(linalg);
Warning, new definition for norm
Warning, new definition for trace

[BlockDiagonal, GramSchmidt, JordanBlock, LUdecomp, QRdecomp,

    Wronskian, addcol, addrow, adj, adjoint, angle, augment, backsub,

    band, basis, bezout, blockmatrix, charmat, charpoly, cholesky,

    col, coldim, colspace, colspan, companion, concat, cond, copyinto,

    crossprod, curl, definite, delcols, delrows, det, diag, diverge,

    dotprod, eigenvals, eigenvalues, eigenvectors, eigenvects,

    entermatrix, equal, exponential, extend, ffgausselim, fibonacci,

    forwardsub, frobenius, gausselim, gaussjord, geneqns, genmatrix,

    grad, hadamard, hermite, hessian, hilbert, htranspose, ihermite,

    indexfunc, innerprod, intbasis, inverse, ismith, issimilar,

    iszero, jacobian, jordan, kernel, laplacian, leastsqrs, linsolve,

    matadd, matrix, minor, minpoly, mulcol, mulrow, multiply, norm,

    normalize, nullspace, orthog, permanent, pivot, potential,

    randmatrix, randvector, rank, ratform, row, rowdim, rowspace,

    rowspan, rref, scalarmul, singularvals, smith, stack, submatrix,

    subvector, sumbasis, swapcol, swaprow, sylvester, toeplitz, trace,

    transpose, vandermonde, vecpotent, vectdim, vector, wronskian]

> alias(Id=&*());

                                I, Id

> f:=proc(i,j)
> if i=j then 0 else 1 fi end;

            f := proc(i, j) if i = j then 0 else 1 fi end

> A:=matrix(7,7,f);

                     [0    1    1    1    1    1    1]
                     [                               ]
                     [1    0    1    1    1    1    1]
                     [                               ]
                     [1    1    0    1    1    1    1]
                     [                               ]
                A := [1    1    1    0    1    1    1]
                     [                               ]
                     [1    1    1    1    0    1    1]
                     [                               ]
                     [1    1    1    1    1    0    1]
                     [                               ]
                     [1    1    1    1    1    1    0]

> evalm(x*Id-A);

               [x     -1    -1    -1    -1    -1    -1]
               [                                      ]
               [-1    x     -1    -1    -1    -1    -1]
               [                                      ]
               [-1    -1    x     -1    -1    -1    -1]
               [                                      ]
               [-1    -1    -1    x     -1    -1    -1]
               [                                      ]
               [-1    -1    -1    -1    x     -1    -1]
               [                                      ]
               [-1    -1    -1    -1    -1    x     -1]
               [                                      ]
               [-1    -1    -1    -1    -1    -1    x ]

> p:=det(");

                              2        3       4       5    7
         p := -6 - 35 x - 84 x  - 105 x  - 70 x  - 21 x  + x

> lambda:=solve(p=0,x);

                 lambda := 6, -1, -1, -1, -1, -1, -1

> factor(p);

                                          6
                           (x - 6) (x + 1)

> ES[1]:=nullspace(lambda[1]*Id-A);

                   ES[1] := {[1, 1, 1, 1, 1, 1, 1]}

> ES[2]:=nullspace(lambda[2]*Id-A);

ES[2] := {[-1, 0, 0, 0, 1, 0, 0], [-1, 0, 0, 0, 0, 0, 1],

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

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

> P:=augment(op(ES[1]),op(ES[2]));

                  [1    -1    -1    -1    -1    -1    -1]
                  [                                     ]
                  [1     0     0     0     1     0     0]
                  [                                     ]
                  [1     0     0     0     0     0     1]
                  [                                     ]
             P := [1     0     0     0     0     1     0]
                  [                                     ]
                  [1     0     1     0     0     0     0]
                  [                                     ]
                  [1     0     0     1     0     0     0]
                  [                                     ]
                  [1     1     0     0     0     0     0]

> evalm(1/P &* A &* P);

               [6     0     0     0     0     0     0]
               [                                     ]
               [0    -1     0     0     0     0     0]
               [                                     ]
               [0     0    -1     0     0     0     0]
               [                                     ]
               [0     0     0    -1     0     0     0]
               [                                     ]
               [0     0     0     0    -1     0     0]
               [                                     ]
               [0     0     0     0     0    -1     0]
               [                                     ]
               [0     0     0     0     0     0    -1]

> 

1. The 5*5 matrix A=[a_ij], with a_ij=i+j-1 if i=j , i+j+1 if i<>j.
2. The 4*4 matrix B=[b_ij], with b_ij=1 if i>j , o otherwise.
3. The 5*5 matrix C=[c_ij], with c_ij=2 if i=j , 1 if i<>=j.

[Last Update: 1999.5.14] Dongseo University Cyber Campus