Originally Posted by

**mivanova** Can you, please help me with this. Even with the Hint I don't know what to do

Show that the following are equivalent

(a) $\displaystyle \begin{bmatrix}I_p&U\\U^*&I_m\end{bmatrix}$ is positive definite;

(b) $\displaystyle I_p - UU^*$ is positive definite;

(c) $\displaystyle I_m - U^*U$ is positive definite;

Hint: Consider $\displaystyle \begin{bmatrix}v^*&w^*\end{bmatrix} \begin{bmatrix}I_p&U\\U^*&I_m\end{bmatrix} \begin{bmatrix}v\\w\end{bmatrix} = v^*(I_p - UU^*)v+{}? = w^*(I_m - U^*U)w+{}?$.