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eigenvalues and eigenvectors of matrix

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I have some very basic lin algerbra eigenvalue problems. (See attached file for full problem description)

1. Find the eigenvalues and eigenvectors for the projection matrix P = [0.2 0.4 0; 0.4 0.8 0; 0 0 1];
2. Find the eigenvalues for the permutation matrix P = [0 1 0; 0 0 1; 1 0 0];
3. Finish the last row to make the matrix A a Markov matrix and find the steady state vector.
4. Compute (A^H)A and A(A^H) for A = [j 1 j; 1 j j].

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The solution includes step-by-step explanations of finding the eigenvalues and eigenvectors of matrix. It also covers two special matrices: Markov matrix and Hermitian matrix.

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