Let $O\in\mathbb{R}^{n\times m}$, $m>n$, be a matrix with orthonormal rows, that is $O O^\top =I_n$, where $\bullet^\top$ denotes transposition and $I_n$ the $n\times n$ identity matrix. Partition $O$ as follows $$ O=\left[O_1 | O_2\right], $$ where $O_1\in\mathbb{R}^{n\times n}$ and $O_2\in\mathbb{R}^{n\times (n-m)}$.
Let $\{\lambda_i(X)\}_{i=1}^n$ denote the set of eigenvalues of a matrix $X\in\mathbb{R}^{n\times n}$ and consider the following subset of the space of row-orthogonal matrices: $$ \mathscr{O}:=\left\{\,O=\left[O_1 | O_2\right]\in \mathbb{R}^{n\times m}, OO^\top =I_n\,:\, |\lambda_i(O_1)|<1,\, i=1,\dots,n\,\,\right\}. $$
My question. Are there other equivalent ways to characterize the subset $\mathscr{O}$ that do not directly involve the eigenvalues of $O_1$?
Any comment, help, suggestion or pointer to the literature is really appreciated. Thanks a lot!
A matrix $O_1$ will work as a block entry if and only if it satisfies $\|O_1\| \leq 1$. For a fixed $O_1$ satisfying $\|O_1\| \leq 1$, it suffices to take any $O_2$ satisfying $O_2 O_2^T = I - O_1O_1^T$.
As I state in the comment: in the "interesting case", we're looking for all matrices $O_1$ satisfying $|\lambda_{max}(O_1)| < \|O_1\| = 1$. To that end, it suffices to apply the criteria discussed on this post.