= Adjoint eigenvector identity for a normal matrix
{title2=$Ax=lambda xRightarrow A^dagger x=overlinelambda x$}
If a <normal matrix> satisfies $Ax=\lambda x$, then $A^\dagger x=\overline\lambda x$. Indeed, $M=A-\lambda I$ is normal and $\|Mx\|^2=x^\dagger M^\dagger Mx=x^\dagger MM^\dagger x=\|M^\dagger x\|^2$. Since $Mx=0$, also $M^\dagger x=0$. This identity proves orthogonality of distinct <eigenspaces> and gives spectral restrictions for <Hermitian matrices>, <skew-Hermitian matrices> and <unitary matrices>.
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