Orthogonal Similarity via Transversal Intersection of Manifolds

Author

Hanfei XuFollow

Author ORCID Identifier

https://orcid.org/0009-0006-2842-0738

Date of Award

8-2024

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics and Statistics

First Advisor

Zhongshan Li

Second Advisor

Marina Arav

Third Advisor

Frank Hall

Fourth Advisor

Hendricus van der Holst

Abstract

Let A be an n × 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(sgn(A)) (consisting of all real matrices having the same sign pattern as A), both considered as embedded smooth submanifolds of R^{n × n}, intersect transversally at A. More specifically, with S = [s_ij ] being the n×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 like OSTP2 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, and we gives several examples illustrating some applications.

Recommended Citation

Xu, Hanfei, "Orthogonal Similarity via Transversal Intersection of Manifolds." Dissertation, Georgia State University, 2024.
https://scholarworks.gsu.edu/math_diss/95

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