I'm being asked to find the spectral decomposition of a hermitian matrix... and I know that if i have 2 distinct eigenvalues, their eigenvectors MUST be normal to each other... thus... I am confused... becasue for this simple matrix...

[1......3+4i]

[3-4i......1]

I am getting the distinct eigenvalues 6 and -4.... and

eigenvectors [(3+4i)/5 1] and [-(3+4i)/5 1]

which are definatley not orthogonal...

what in blazes am I doing wrong here

going crrrrrrrrrrazy!