Hey again! So I got intimate with Schur complements over the weekend and have come to the following conclusions.

First, for the sake of clarity and aid, these are the important inequalities for the shur matrices in the \mathop{\textrm{Tr}}(S^{-2})\leq t case (only final answer is shown):

for the first matrix (the matrix with the S in the bottom right), the important inequality is X \geq S^{-1}

for the other matrix, the important inequality is Z \geq X^2

Note1: Seems understandable to me!

Note2: There is a statement in the original link that also confirms my findings.

Second, the case you provided, for \mathop{\textrm{Tr}}((G^TG)^{-1})\leq t:

for the first matrix (the matrix with G in it), the important inequality is X \geq GG^T

for the other matrix, the important inequality is Z \geq X^2

## Note3: It seems to me that the first matrix needs G and G^T switched.

Note4: It seems to me that the second matrix achieves an inequality that is not needed in this case.

Third, is a case where I made the matrices according to how I see fit (learning from the first two cases), here is what I have:

$$\begin{bmatrix} X & G^T \G & I \end{bmatrix} \succeq 0 \qquad \begin{bmatrix} Z & I \ I & X \end{bmatrix} \succeq 0 \qquad \mathop{\textrm{Tr}}(Z)\leq t$$

for the first matrix (the matrix with G in it), the important inequality is X \geq G^TG

for the other matrix, the important inequality is Z \geq X^{-1}

Note5: I’m not so sure about this, still thinking about it…

So, I am currently not with my desktop computer to stick this in CVX and see what happens, I will be doing that later today. In the mean time, I am trying to familiarize myself with the math. Any pointers, comments, and remarks are greatly appreciated.

Thank you for reading : )