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A Hermitian operator A has the spectral decomposition

A = Σ a

_{n}|n><n| (summation in n)

where A|n> = a

_{n}|n> (the a

_{n}'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

AA

^{-1}= A^{-1}A = 1?My intuition would be A = Σ (1/a

_{n})|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!