Matrix inverse and Hermitian transpose

Question

Could anyone help me to prove the following the equation?

$\large ( G_2^H G_2 + K_w^{-1} )^{-1} = Q$

which leads to

$\large K_w = Q - Q G_2^H ( G_2 Q G_2^H - I )^{-1} G_2 Q$

Here $A^H$ is the Hermitian transpose (conjugate transpose) of $A$.

Answer 1

Seems to me a simple case of the Woodbury matrix identity: $$ (A+UCV)^{-1}=A^{-1}-A^{-1}U(C^{-1}+UA^{-1}V)^{-1} VA^{-1}$$ For more details see:

http://en.wikipedia.org/wiki/Woodbury_matrix_identity