I am trying to find a relation between $U$ and $V$, such that U is a $(p\times d)$ semi-orthogonal matrix (i.e $\ U^{T}U=\mathbf{I}_{d}$) and $V$ is a $(p\times (p-d))$ matrix which is the completion of U. Such that, $\Gamma=[U \ ,\ V]$ is an orthogonal $(p\times p)$ matrix (i.e $\ U^{T}U=\mathbf{I}_{d}= \Gamma\Gamma^T$).
My working:
$\Gamma^T\Gamma =[U^T V^T]^T\ [U \ ,\ V]=\mathbf{I}_{p\times p}$
Therefore:
$\\\bigg[\begin{matrix}U^TU & U^TV\\V^TU &V^TV\end{matrix}\bigg]=\bigg[\begin{matrix}\mathbf{I}_{d\times d} & \mathbf{0}_{d\times p-d}\\\mathbf{0}^T_{d\times p-d} & \mathbf{I}_{p-d\times p-d} \end{matrix}\bigg] $ Which implies that $U^TV=\mathbf{0}_{d\times p-d}$ and $V^TU=\mathbf{0}^T_{d\times p-d}$ and that V is also a semi-orthogonal matrix.
I also have the following equation in terms of U and V: $\det(A)\det(U^T(A)^{-1}U)=\det(V^TAV)$ for any $p\times p$ symmetric matrix A.
My Working:
$\det(A)\det(U^T(A)^{-1}U)=\det(V^TAV)$
I am quite sure that there should be a relation but I cannot see it yet. Thanks in advance.