Woodbury Matrix Identity - Direct Proof

Direct Proof

Just check that times the RHS of the Woodbury identity gives the identity matrix:

\begin{align} &\left(A+UCV \right) \left \\ & \quad = I + UCVA^{-1} - (U+UCVA^{-1}U)(C^{-1} + VA^{-1}U)^{-1}VA^{-1} \\ & \quad = I + UCVA^{-1} - UC(C^{-1}+ VA^{-1}U)(C^{-1} + VA^{-1}U)^{-1}VA^{-1} \\ & \quad = I + UCVA^{-1} - UCVA^{-1} = I \end{align}

Read more about this topic:  Woodbury Matrix Identity

Famous quotes containing the words direct and/or proof:

    “Long as I have lived, and many blasphemers as I have heard and seen, I have never yet heard or witnessed any direct and conscious blasphemy or irreverence; but of indirect and habitual, enough. Where is the man who is guilty of direct and personal insolence to Him that made him?”
    —Henry David Thoreau (1817–1862)

    “The thing with Catholicism, the same as all religions, is that it teaches what should be, which seems rather incorrect. This is “what should be.” Now, if you’re taught to live up to a “what should be” that never existed—only an occult superstition, no proof of this “should be”Mthen you can sit on a jury and indict easily, you can cast the first stone, you can burn Adolf Eichmann, like that!”
    —Lenny Bruce (1925–1966)