Suppose we have real matrices $P,Q$. I was wondering if the eigenvalues of $[PP'\quad PQ';\\ QP' \quad QQ']$ could be related to the eigen values of $PP'+QQ'$? where $P'$ refers to the standard transpose.
Are there specific conditions on $P,Q$ based on which I can prove the eigen values of the block matrix are related to $PP'+QQ'$? Maybe, can we say, the the trace norm of the block matrix is at most the trace norm of $PP'+QQ'$?
Let $M = \begin{bmatrix} P \\ Q \end{bmatrix}$. Then, we have $$ MM' = \begin{bmatrix} PP' & PQ' \\ QP' & QQ' \end{bmatrix}$$ and $$ M'M = PP' + QQ'. $$ That is, you asked whether $MM'$ and $M'M$ have the same set of eigenvalues.
In general that is not true, as $0$ is always an eigenvalue of $MM'$, whereas $M'M$ might be positive definite. For example consider $$ \begin{bmatrix} 1 \\ 0 \end{bmatrix}. $$
However, $MM'$ and $M'M$ do have the same nonzero eigenvalues. Let $M=VSU'$ be the singular value decomposition of $M$. That is, $U$ and $V$ are unitary and $S$ is of the same size as $M$ and diagonal with the singular values on the diagonal. Then, we have $$ MM' = VSS'V' $$ and $$ M'M = US'SU'. $$ Now, $SS'$ and $S'S$ are square diagonal matrices and their nonzero diagonal elements are the same, namely the squared singular values of $M$.