Orthogonal similarity via transversal intersection of manifolds

Let $A$ be an $n\times n$ real matrix. We define orthogonal similarity-transversality property (OSTP) if the smooth manifold consisting of the real matrices orthogonally similar to $A$ and the smooth manifold $Q(\text{sgn}(A))$ (consisting of all real matrices having the same sign pattern as $A$), both considered as embedded smooth submanifolds of $\mathbb R^{n \times n}$, intersect transversally at $A$. More specifically, with $S=[s_{ij}]$ being the $n\times n$ generic skew-symmetric matrix whose strictly lower (or upper) triangular entries are regarded as independent free variables, we say that $A$ has the OSTP if the Jacobian matrix of the entries of $AS-SA$ at the zero entry positions of $A$ with respect to the strictly lower (or upper) triangular entries of $S$ has full row rank. We also formulate several properties and show that if a matrix $A$ has the OSTP, then every superpattern of the sign pattern sgn$(A)$ allows a matrix orthogonally similar to $A$, and every matrix sufficiently close to $A$ also has the OSTP. This approach provides a theoretical foundation for constructing matrices orthogonally similar to a given matrix while the entries have certain desired signs or zero-nonzero restrictions. In particular, several important classes of zero-nonzero patterns and sign patterns that require or allow the OSTP are identified. We provide several Examples illustrating some applications.

Last modified: Thu Apr 11 10:59:49 EDT 2024