= Kitaev geometrical lemma
{c}
{title2=$\lambda_{\min}(A+B)\geq2\gamma\sin^2(\vartheta/2)$}
= Kitaev's geometrical lemma
{c}
{synonym}
Suppose <positive operators> $A,B$ have positive <eigenvalues> at least $\gamma$ and their nullspaces meet only at zero. If $\vartheta$ is their <smallest angle between two subspaces>, then $A+B\geq2\gamma\sin^2(\vartheta/2)I$. Indeed $A+B\geq\gamma(2I-P-Q)$, while $(P+Q)^2\leq(1+\cos\vartheta)(P+Q)$ gives $P+Q\leq(1+\cos\vartheta)I$. With a common nullspace, apply the same proof on its orthogonal complement to bound the positive <spectral gap>.
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