Another quantum question.....I feel dumb . A Hermitian operator A has the spectral decomposition A = Σ an|n><n| (summation in n) where A|n> = an|n> (the an's are eigenvalues of A, and |n>'s the eigenstates). So, how can I find the spectral decomposition of the inverse of A so that AA-1 = A-1A = 1? My intuition would be A = Σ (1/an)|n><n| (summation in n), since 1 = Σ |k><k| (summation in k), but I don't think it's that easy. Thanks in advance!