Consider $y = X\beta + \epsilon$, where $X : n \times p $ matrix with column rank $r < p$ and $\beta = (\beta_1 , \dots, \beta_p)^T$. Let $C: m\times p$ be a rank $m$ matrix of constants such that $C\beta$ is estimable. Let $\hat{\beta}_C$ be a least-squares estimator of $\beta$ subject to the restriction that $\beta$ satisfies $C\beta=0$. Let $U$ be any matrix whose columns form a basis for the null space of $C^T$. Show that $\hat{\beta}_C = U(U^TX^TXU)^- U^TX^Ty$.
I use the lagrange multiplier to get $\hat{\beta}_ C=(I-B)\hat{\beta}$, where $\hat{\beta}= (X^TX)^-X^Ty$, and $B= (X^TX)^-C^T[C(X^TX)^-C^T]^{-1}C$.
So I tried to show that $(I-B) = U(U^TX^TXU)^-X^T$, but can't get the result what I want.
How could I solve this problem?