Statistics: Sherman-Morrison-Woodbury Formula

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Sherman-Morrison-Woodbury Formula

If A − 1 and D − 1 exist then \left( A + U D V \right)^{-1} = A^{-1}-A^{-1}U \left( D^{-1}+VA^{-1}U \right)^{-1}VA^{-1}.

Sherman-Morrison-Woodbury formula and proof

Related formulas


V \left( A + U D V \right)^{-1} = V A^{-1}-VA^{-1}U \left( D^{-1}+VA^{-1}U \right)^{-1}VA^{-1}
=\left( I-VA^{-1}U \left( D^{-1}+VA^{-1}U \right)^{-1} \right) VA^{-1}
=\left( D^{-1}+VA^{-1}U -VA^{-1}U \right) \left( D^{-1}+VA^{-1}U \right)^{-1}  VA^{-1}
=D^{-1} \left( D^{-1}+VA^{-1}U \right)^{-1}  VA^{-1}


\left( A + U D V \right)^{-1}U = A^{-1}U \left( D^{-1}+VA^{-1}U \right)^{-1}D^{-1}