> 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 i+j-1 else i+j+1 fi end;

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

> A:=matrix(4,4,f);

                            [1    4    5    6]
                            [                ]
                            [4    3    6    7]
                       A := [                ]
                            [5    6    5    8]
                            [                ]
                            [6    7    8    7]

> evalm(x*Id-A);

                [-1 + x      -4        -5        -6  ]
                [                                    ]
                [  -4      -3 + x      -6        -7  ]
                [                                    ]
                [  -5        -6      -5 + x      -8  ]
                [                                    ]
                [  -6        -7        -8      -7 + x]

> p:=det(");

                                        2       3    4
               p := -256 - 336 x - 140 x  - 16 x  + x

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

                                1/2           1/2
             lambda := 10 + 2 41   , 10 - 2 41   , -2, -2

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

     [              1/2                1/2                  1/2]
    {[- 5/2 + 1/2 41   , - 3/4 + 1/4 41   , 1, 11/4 - 1/4 41   ]}

> factor(p);

                        2                     2
                      (x  - 20 x - 64) (x + 2)

> 

1. Find all the eigenvalues and eigenvectors of the 9*9 matrix A=[a_ij] with a_ij=1 for all i and j.
2. Find all the eigenvalues and eigenvectors of the 4*4 matrix B=[b_ij], where, for each i and j, b_ij is the minimum of i and j.
3. Find all the eigenvalues and eigenvectors of the 3*3 matrix C=[c_ij], where, for each i and j, c_ij=i/j.

[Last Update: 1999.4.16] Dongseo University Cyber Campus