Another quantum question.....I feel dumb .(adsbygoogle = window.adsbygoogle || []).push({});

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^{-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!

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Need help! Inverse of an operator

**Physics Forums | Science Articles, Homework Help, Discussion**