Transforming a matrix of explanatory variable in regression
$\boldsymbol X^*_1$ is the residual of the projection of $\boldsymbol X_1 $ on $\boldsymbol X_2 $. The 'residual maker' matrix is $\boldsymbol I - \boldsymbol X_2 \left(\boldsymbol X_2^T\boldsymbol X_2\right)^{-1}\boldsymbol X_2^T$. Therefore, $$ \boldsymbol X_1^* = \left(\boldsymbol I -\boldsymbol X_2 \left(\boldsymbol X_2^T\boldsymbol X_2\right)^{-1}\boldsymbol X_2^T\right)\boldsymbol X_1.$$