Matrix inversion lemma¶
The matrix inversion lehma for a patritioned matrix:
\[\begin{split} M = \begin{bmatrix} E & F \\ G & H \end{bmatrix}
\end{split}\]
The inverse is:
\[\begin{split} M^{-1} = \begin{bmatrix} (M/H)^{-1} & -(M/H)^{-1} FH^{-1} \\ -H^{-1}G(M/H)^{-1} & H^{-1} + H^{-1}G(M/H)^{-1}FH^{-1} \end{bmatrix}\end{split}\]
\(M/H = E - FH^{-1} G\) is the Schur Complement of M with respect to H
We can express the inverse only in terms of \(E\) and \(M/E\)
\[\begin{split} M^{-1} = \begin{bmatrix} E^{-1} + E^{-1}F(M/E)^{-1}GE^{-1} & -E^{-1}F(M/E)^{-1} \\ -(M/E)^{-1}GE^{-1} & (M/E)^{-1} \end{bmatrix} \end{split}\]
\(M/E = H - GE^{-1}F\) is the Schur Complement of M with respect to E
From these we can derive the Matrix inversion lemma or Sherman-Morrision-Woodbury formular:
\[1.(E - FH^{-1}G)^{-1} = E^{-1} + E^{-1}F(H-GE^{-1}F)^{-1}GE^{-1}\]
\[2.(E - FH^{-1}G)^{-1}FH^{-1} = E^{-1}F(H - GE^{-1}F)^{-1}\]
And the Matrix determinant lemma
\[3. |E - FH^{-1}G| = |H-GE^{-1}F||H^{-1}||E|\]
Application¶
We can perform an symmetric rank one update to an inverse matrix:
\(H = -1; F = u; G = v^T\) then:
\[\begin{split}
(E + uv^T)^{-1} = E^{-1} + E^{-1}u(-1 - v^TE^{-1}u)^{-1} v^TE^{-1} \\ = E^{-1} - \frac{E^{-1} uv^T E^{-1}}{1 + v^TE^{-1}u}
\end{split}\]
