positive definite matrix as a matrix blocks

Question

If $X$ is a rectangular matrix of $m \times n$, is it true that

$$ \begin{bmatrix} UU' & X\\ X' & VV' \end{bmatrix} $$ is a PSD ? where $UU'$ is $n \times n$ and $VV'$ is $m \times m$.

When I try to work out the condition for PSD, $y'Zy$, I get

$$ y'UU'y + y'VV'y + y'(X' + X)y $$ clearly first two terms are PSD, but the third term is not even defined.

How can I show that third term is also PSD ?

Answer 1

It is not true.

Consider $\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$ with determinant $-1$.