Proof that $\hat{\beta^{t}}X^{t}y = \hat{y}^{t}\hat{y}$

Question

I need to prove that:

$\hat{\beta^{t}}X^{t}y = \hat{y}^{t}\hat{y}$

I know that:

$\hat{\beta}= \left ( X^{t}X \right )^{-1}X^{t}y$

$\hat{y}= X\left ( X^{t}X \right )^{-1}X^{t}y= X\hat{\beta}=Hy$

Here's what I have so far:

$\hat{y}^{t}\hat{y} = \left ( X\hat{\beta} \right )^{t}X\left ( X^{t}X \right )^{-1}X^{t}y= \hat{\beta}^{t}X^{t}Hy$

I'm clearly doing something wrong, any idea what I can change to prove it?

Answer 1

You almost have it. From your last step:

$$ \hat{y}^{t}\hat{y} = \left ( X\hat{\beta} \right )^{t}X\left ( X^{t}X \right )^{-1}X^{t}y $$

$$ = \hat{\beta}^tX^tX\left ( X^{t}X \right )^{-1}X^{t}y $$

$$ = \hat{\beta}^tX^{t}y . $$