Does there exist an analytical characterization of the set of matrices $\Gamma_k\in\mathbb{R}^{m\times n}$ such that both
$$ \sum_{k=1}^K\Gamma_k^T\Lambda_0\Gamma_k=I $$ and $$ \sum_{k=1}^K\Gamma_k\Lambda_1\Gamma_k^T=I $$ hold, for a particular choice of real diagonal matrices $\Lambda_i$ and $I$ being the identity matrix?
For $\Lambda_0=I$, the first condition holds if $$ U_0=\begin{pmatrix} \Gamma_1 \\ \vdots\\ \Gamma_K\end{pmatrix} $$ is a column-orthogonal matrix, and similarly, the second condition holds for $\Lambda_1=I$ if $$ U_1=\begin{pmatrix} \Gamma_1^T \\ \vdots\\ \Gamma_K^T\end{pmatrix} $$ is column-orthogonal.
For context, this is of interest when constructing canonical representations of so-called matrix product states.