How Do I find Eigen Vectors?

In Maple, Mathematica, and Matlab, how do I obtain eigenvalues and eigenvectors?

Maple

In Maple, the command to obtain eigenvalues and eigenvectors is:

  eigenvects( map(evalf, A) )

Example:

  > with(linalg):  
  > A := matrix( [ [10, 0, 1], [0, 5, 1], [1, 1, 3] ] );

  [ 10   0  1 ]
  [           ]
                                 A := [  0   5  1 ]
  [           ]
  [  1   1  3 ]

  > eigsys := eigenvects( map(evalf, A) );

  eigsys := 
     [2.471653325, 1, { [-.1225891913, -.3650187460, .9228939292]} ],
     [10.14389544, 1, { [ .9894258912, .02767822018, .1423738801]} ],
     [5.384451215, 1, { [.07751319726, -.9305886462,-.3577659480]} ]

  > eigsys[2][3][1][3];

  .1423738801

Mathematica

In Mathematica, use:

  {d,v} = Eigensystem[N[a]]

Example:

  In[1]:= a = {{10., 0, 1}, {0, 5, 1}, {1, 1, 3}}

  Out[1]= {{10., 0, 1}, {0, 5, 1}, {1, 1, 3}}

  In[2]:= eigsys = Eigensystem[N[a]]

  Out[2]= { {10.1439, 5.38445, 2.47165},
  >         { {-0.989426,-0.0276782, -0.142374},
  >           {0.0775132, -0.930589, -0.357766},
  >           { 0.122589,  0.365019, -0.922894} } }

  In[3]:= eigsys[[2,1,3]]

  Out[3]= -0.142374

Matlab

In Matlab, use:

  [V,D] = eig(A)

Example:

  >> A = [ 10., 0, 1; 0, 5, 1; 1, 1, 3 ]

  A =

  10     0     1
   0     5     1
   1     1     3

  >> [V,D] = eig(A);
  >> [V,D] = eig(A)

  V =

  0.9894    0.0775   -0.1226
  0.0277   -0.9306   -0.3650
  0.1424   -0.3578    0.9229

  D =

  10.1439         0         0
        0    5.3845         0
        0         0    2.4717

  >> V(3,1)

  ans =

  0.1424