Matrices Eigenvalues - sympy/sympy GitHub Wiki

    >>> from sympy.matrices import Matrix
    >>> M = Matrix(4, 4, lambda i,j: i*j+1)
    >>> M.eigenvals()                           #doctest: -PRETTY, +RELEASE_ONLY
    {0: 2, -61**(1/2) + 9: 1, 61**(1/2) + 9: 1}
    >>> M.eigenvals()                           #doctest: -PRETTY, +FUTURE_ONLY
    {0: 2, -sqrt(61) + 9: 1, sqrt(61) + 9: 1}

    >>> M = Matrix(4, 4, lambda i,j: i*j+2)
    >>> M.eigenvals()
    {0: 2, 2: 1, 20: 1}

    >>> M = Matrix(4, 4, lambda i,j: i*j+3)
    >>> M.eigenvals()
               _____            _____         
    {0: 2, - \/ 109  + 13: 1, \/ 109  + 13: 1}
    >>> M
    [3  3  3  3 ]
    [           ]
    [3  4  5  6 ]
    [           ]
    [3  5  7  9 ]
    [           ]
    [3  6  9  12]

 >>> from sympy.abc import x
 >>> M = Matrix(4, 4, lambda i,j: i*j+x)
 >>> M
    [x    x      x      x  ]
    [                      ]
    [x  x + 1  x + 2  x + 3]
    [                      ]
    [x  x + 2  x + 4  x + 6]
    [                      ]
    [x  x + 3  x + 6  x + 9]

 >>> M.eigenvals()
                    _________________                  _________________        
                   /    2                             /    2                    
    {0: 2, 2*x - \/  4*x  + 8*x + 49  + 7: 1, 2*x + \/  4*x  + 8*x + 49  + 7: 1}

Such things are really tedious to do by hand, but in SymPy one can do them just fine.