1. ## posititve definite

Show that the following are equivalent
(a) ( In U is posititve definite;
U* Im)
(b) Ip - UU* is positive definite;
(c) Im - UU* is positive definite;
Hint: Consider
(v*, w*)( Ip U ) (v w)' = v*(Ip - UU*)v+?= w*(Im - UU*)w+?
U* Im)
Thank you!

2. 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 $\displaystyle \begin{bmatrix}v^*&w^*\end{bmatrix} \begin{bmatrix}I_p&U\\U^*&I_m\end{bmatrix} \begin{bmatrix}v\\w\end{bmatrix} = v^*(I_p - UU^*)v+{}? = w^*(I_m - U^*U)w+{}?$.
First, note that I have made some corrections to the question. The most important one is that some of the UU*s should be U*U.

Start by doing what the Hint suggests, and multiply the matrices to get the product

$\displaystyle \begin{bmatrix}v^*&w^*\end{bmatrix} \begin{bmatrix}I_p&U\\U^*&I_m\end{bmatrix} \begin{bmatrix}v\\w\end{bmatrix} = \begin{bmatrix}v^*&w^*\end{bmatrix} \begin{bmatrix}v+Uw\\U^*v+w\end{bmatrix} = v^*v+v^*Uw + w^*U^*v + w^*w$.

Now comes the clever part. The Hint suggests that you should write this in the form $\displaystyle v^*(I_p - UU^*)v+{}?$. But $\displaystyle v^*(I_p - UU^*)v = v^*v - v^*UU^*v = v^*v - (U^*v)^*(U^*v)$ (because $\displaystyle (U^*v)^* = v^*U$). So we get

\displaystyle \begin{aligned}\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 + (U^*v)^*(U^*v) + (v^*U)w + w^*(U^*v) + w^*w \\ &= v^*(I_p - UU^*)v + (U^*v+w)^*(U^*v+w).\end{aligned}

In the equation $\displaystyle \begin{bmatrix}v^*&w^*\end{bmatrix} \begin{bmatrix}I_p&U\\U^*&I_m\end{bmatrix} \begin{bmatrix}v\\w\end{bmatrix} = v^*(I_p - UU^*)v + (U^*v+w)^*(U^*v+w)$, you can see that if $\displaystyle I_p - UU^*$ is positive definite then the right side of the equation will be positive for all v and w. Therefore the left side is positive, which means that the matrix $\displaystyle \begin{bmatrix}I_p&U\\U^*&I_m\end{bmatrix}$ is positive definite. On the other hand, if the matrix is positive definite then the left side of the equation is positive for all v and w, and in particular when $\displaystyle w = -U^*v$. That implies that $\displaystyle v^*(I_p - UU^*)v$ is positive for all v, which tells us that $\displaystyle I_p - UU^*$ is positive definite.

Thus condition (a) is equivalent to condition (b). A similar calculation shows that (a) is equivalent to (c).