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:

    “The frequency of personal questions grows in direct proportion to your increasing girth. . . . No one would ask a man such a personally invasive question as “Is your wife having natural childbirth or is she planning to be knocked out?” But someone might ask that of you. No matter how much you wish for privacy, your pregnancy is a public event to which everyone feels invited.”
    —Jean Marzollo (20th century)

    “War is a beastly business, it is true, but one proof we are human is our ability to learn, even from it, how better to exist.”
    —M.F.K. Fisher (1908–1992)