Author ORCID Identifier
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 Rn × n, intersect transversally at A. More specifically, with S = [sij ] 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.
DOI
https://doi.org/10.57709/37369939
Recommended Citation
Xu, Hanfei, "Orthogonal Similarity via Transversal Intersection of Manifolds." Dissertation, Georgia State University, 2024.
doi: https://doi.org/10.57709/37369939
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