0
$\begingroup$

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 ?

$\endgroup$
  • $\begingroup$ 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. $\endgroup$ – Laray Apr 10 '18 at 9:47
0
$\begingroup$

It is not true.

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

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.