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I believe this result is due to Stewart, but I haven't been able to track it down: let $A$ have full column rank and let $B = A + E$ where $P_{A} E = 0$. Then $$ \|(I - P_B)P_A\|^2_2 = 1 - \lambda_{\text{min}}(A(B^TB)^{-1} A^T) $$

Any idea where it's stated? And any extensions?

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4 
 What are $P_A$ and $P_B$? – Stanislav Jun 8 2012 at 17:43
$P_A$ is the projection onto the range space of $A.$ – Alex Gittens Jun 13 2012 at 15:57

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