Finding eigenvalues and eigenvectors for projection matrix

Question

Problems:

Eigenvalues and eigenvectors of matrix

1. Find the eigenvalues and eigenvectors for the projection matrix

                    P = [0.2 0.4 0; 0.4 0.8 0; 0 0 1]

                   P has λ = 1,0,1 with eigenvectors ( 1,2,0 ), (2, -1,0 ), (0, 0, 1)

2. Find the eigenvalues for the permutation matrix

                        P = [0 1 0; 0 0 1; 1 0 0];

det(P- λI) = 0 gives the equation λ³ = 1. This reflects the fact that P3 = I. The solutions of λ³ = 1 are λ = 1 (real) and λ = e^2xi/3 ( complex conjugates). The real eigenvector x₁ = ( 1,1,1 ) is not changed by the permutationP. The complex eigenvectors are x₂ = (1, e^-2xi/3,e^-4xi/3)
and x₃ = ( 1, e^2xi/3,e^-4xi/3) = x^-₂

3. Finish the last row to make the matrix A a Markov matrix and find the steady state vector.

                    A = [.7.1.2; .1.6.3; _ _ _]

4. Compute A^HA and AA^H for A = [i 1 i: 1 i i]

           A^HA = [ 2     0     1+i] and AA^H = [ 3   1 ] are hermitian matrices.

                      [ 0     2     1+i]                  [ 1   3 ]

                      [1-i   1-i       2]