Construct a complex Hermitian matrix by setting
j = sqrt(-1); A = rand(5) + j * rand(5); A = (A + A')/2
(a) The eigenvalues of A should be real. Why? Compute the eigenvalues and examine your results using format long. Are the computed eigenvalues real? Compute also the eigenvectors by setting
[X, D] = eig(A)
What type of matrix would you expect X to be? Use the MATLAB command X' * X to compute XHX. Do the results agree with your expectations?
(b) Set
E = D + j * eye(5) and B = X * E/X
What type of matrix would you expect B to be? Use MATLAB to compute BHB and BBH. How do these two matrices compare?