> with(LinearAlgebra):
> A := Matrix([[4,-2],[1,1]]);

                                 [4    -2]
                            A := [       ]
                                 [1     1]

> Eigenvalues(A);

                                 [3]
                                 [ ]
                                 [2]

> Eigenvectors(A);

                            [2]  [1    2]
                            [ ], [      ]
                            [3]  [1    1]

> Lambda2 := DiagonalMatrix([lambda,lambda]);

                               [lambda      0   ]
                    Lambda2 := [                ]
                               [  0       lambda]

> charmatA := A - Lambda2;

                            [4 - lambda        -2    ]
                charmatA := [                        ]
                            [    1         1 - lambda]

> charpolyA := Determinant(charmatA);

                                                   2
                 charpolyA := 6 - 5 lambda + lambda

> CharacteristicPolynomial(A,mu);

                                         2
                            6 - 5 mu + mu

> solve(charpolyA = 0);

                                 3, 2

> NullSpace(subs(lambda=2,charmatA));

                                 [1]
                                {[ ]}
                                 [1]

> A2 := matrix(2,2,[0,0,0,0]);

                                  [0    0]
                            A2 := [      ]
                                  [0    0]

> eigenvectors(A2);

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

> A3 := matrix(2,2,[1,2,-2,1]);

                                 [ 1    2]
                           A3 := [       ]
                                 [-2    1]

> eigenvectors(A3);

           [1 + 2 I, 1, {[1, I]}], [1 - 2 I, 1, {[1, -I]}]

> eigenvectors(matrix(2,2,[3,4,3,2]));

                [-1, 1, {[-1, 1]}], [6, 1, {[4/3, 1]}]