Push-through identity
= Push-through identity
For a rectangular matrix $X$ and $\lambda>0$,
$$
X^T(XX^T+\lambda I)^{-1}=(X^TX+\lambda I)^{-1}X^T.
$$
Multiplying either side by $XX^T+\lambda I$ verifies the identity.
= Push-through identity
For a rectangular matrix $X$ and $\lambda>0$,
$$
X^T(XX^T+\lambda I)^{-1}=(X^TX+\lambda I)^{-1}X^T.
$$
Multiplying either side by $XX^T+\lambda I$ verifies the identity.