The rank-one inverse update

ProofGraduate⏱ 1 hourQuantizationblock inverseschur complementlinear algebra

Problem

Prove that for $H\succ0$ with coordinate $q$ removed,

(H_{-q})^{-1} = \Big[H^{-1} - \tfrac{1}{[H^{-1}]_{qq}}H^{-1}e_qe_q^\top H^{-1}\Big]_{-q},

and explain why this makes a full sweep cost one inversion rather than $d$ .

Write a complete argument. Every inequality you use should be one you could state precisely.

Hints — 0 of 2 revealed

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