Let's say we have two row-orthonormal complex matrices $P \in \mathbb{C}^{n \times k}$ and $Q \in \mathbb{C}^{m \times k}$, $n + m < k$. Since those martices are row-orthonormal, rows of $P$ form an orthonormal basis in $\mathbb{C}^n$, and rows of $Q$ form an orthonormal basis in $\mathbb{C}^m$.
I want to find an orthonormal basis of the union space of $P$ and $Q$, i.e. a space spanned by the rows of matrix $ W = \left[ {\begin{array}{c} P \\ Q \end{array}} \right]$. I know that I can do the orthogonalization of $W$ using the Gram-Shmidt process. But is there a way to do it in a faster way, using the fact that both $P$ and $Q$ are already row-orthonormal?