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- In a maths course the following statement is claimed to be self-evident, and I don't find it so.

We have a matrix ##M = \ket{\psi^{\perp}}\bra{\psi^{\perp}} + \ket{\varphi^{\perp}}\bra{\varphi^{\perp}}##

The claim is that the eigenvalues of such a matrix are ##\lambda_{\pm}= 1\pm |\bra{\psi}\ket{\varphi}|##

Can someone proof this claim? I have been told it is self-evident but I've been already a couple of days struggling with it.

The claim is that the eigenvalues of such a matrix are ##\lambda_{\pm}= 1\pm |\bra{\psi}\ket{\varphi}|##

Can someone proof this claim? I have been told it is self-evident but I've been already a couple of days struggling with it.