The Woodbury matrix identity is defined as follows: $$ {(A+UCV)}^{-1}=A^{-1}-A^{-1}U{(C^{-1}+VA^{-1}U)}^{-1}VA^{-1} $$
I want to use the Woodbury matrix identity theorem to change the following matrix formula $$ W={(XX^T+\lambda G)}^{-1}XY $$ into the following form $$ W=G^{-1} X {(X^TG^{-1}X+\lambda I)}^{-1}Y $$ The dimensions are as follows: $$ X\in R^{p\times n}\\ G\in R^{p\times p}\\ Y\in R^{n\times c} $$ Could anyone help give some hints?
UPDATE:
From the two formulas about $W$, we could get the following equations, thus the two $W$s should be equal: