= Commuting symmetric matrices have a common orthonormal eigenbasis
For commuting real <symmetric matrices> $A,B$, each <eigenspace> of $A$ is invariant under $B$: $A(Bv)=B(Av)=\lambda Bv$. The restriction of $B$ to that <eigenspace> is again symmetric. The <spectral theorem> therefore supplies an <orthonormal eigenbasis> for $B$ inside each <eigenspace> of $A$. Combining these <bases> gives a common <orthonormal eigenbasis> of the whole space. Conversely, two <symmetric matrices> diagonal in the same <orthonormal basis> <commute>.
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