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 $n\times n$ real matrix. As shown in the recent paper \cite{Fallat22}, if the manifolds $\mathcal{M}_A$$=\{ G^{-1}AG: G\in\text{GL}(n,\mathbb{R})\}$ and $Q($sgn$(A))$ (consisting of all real matrices having the same sign pattern as $A)$, both considered as embedded submanifolds of $\mathbb{R}^{n\times n}$, intersect transversally at $A$, then every superpattern of sgn$(A)$ also allows a matrix similar to $A.$ Such paper introduced a condition on $A$ equivalent to the above transversality, called the \emph{nonsymmetric strong spectral property} (nSSP).

In this dissertation, this transversality property of $A$ is characterized using an alternative, more direct and convenient condition, called the \emph{similarity-transversality property} (STP). Let $X=[x_{ij}]$ be a generic matrix of order $n$ whose entries are independent variables. The STP of $A$ is defned as the full row rank property of the Jacobian matrix of the entries of $AX-XA$ at the zero entry positions of $A$ with respect to the nondiagonal entries of $X.$ This new approach makes it possible to take better advantage of the combinatorial structure of the matrix $A$, and provides theoretical foundation for constructing matrices similar to a given matrix while the entries have certain desired signs. In particular, several important classes of zero-nonzero patterns and sign patterns that require or allow this transversality property are identified. Examples illustrating many possible applications are provided.

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