Let $O\in\mathbb{R}^{n\times m}$, $m>n$, be such that $O O^\top =I_n$. (Here $\bullet^\top$ denotes transposition and $I_n$ the $n\times n$ identity matrix.) Consider the following partition of $O$, $$ O=\left[O_1 | O_2\right], $$ where $O_1\in\mathbb{R}^{n\times n}$ and $O_2\in\mathbb{R}^{n\times (n-m)}$. It is easy to show that all the eigenvalues of $O_1$ lie in the closed unit disk of the complex plane.

I'm interested in the subclass of row-orthogonal $O$'s featuring a square block $O_1$ that has all its eigenvalues strictly inside the unit disk (i.e., the modulus of each eigenvalue of $O_1$ is strictly smaller than one). So, my question:

Are there other equivalent ways to characterize such subclass of row-orthogonal matrices?

More precisely, I'm looking for characterizations that do not directly involve the spectrum of $O_1$.


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