# positive definite matrix as a matrix blocks

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 ?

• You have to apply the upper part of $y$ to the upper blocks, and the lower part of $y$ to the lower blocks. That will take care of your $X'+X$ problem. – Laray Apr 10 '18 at 9:47

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