Speaker
Description
The graph ${\mathcal G}(A)$ of a real symmetric matrix $n\times n$ matrix $A=[a_{ij}]$ has vertices $V=\{1,\dots,n\}$ and edges $E=\{ \{i,j\}: a_{ij}\ne 0\mbox{ and } i\ne j\}$. The set of matrices described by a graph $G$ is ${\mathcal S}(G)=\{A\in{\mathbb R}^{n\times n}:{\mathcal G}(A)=G \mbox{ and } A^\top=A\}$ and the
Inverse Eigenvalue Problem of $G$ (IEP-$G$) is to determine all possible spectra (i.e., multisets of eigenvalues) of matrices $A\in {\mathcal S}(G)$. Other inverse eigenvalue problems for $G$ have also been studied based on matrix properties, e.g., the PSD Inverse Eigenvalue Problem of $G$ where matrices must be positive semidefinite.
A symplectic matrix is a $(2n) \times (2n)$ real matrix $S$ such that $S^\top \Omega S=\Omega$ where
$\Omega=\left[\begin{array}{r|r} O&I\\\hline -I&O \end{array} \right].$ It is known that for any $(2n) \times (2n)$ symmetric positive definite matrix $A$, there exists a symplectic matrix $S$ and an $n\times n$ diagonal matrix $D$ such that $S^\top A S= \left[\begin{array}{c|c}D&O\\ \hline
O&D\end{array} \right] .$ The diagonal entries of $D$ are unique (up to re-ordering) and are called the symplectic eigenvalues of $A$; the symplectic spectrum of $A$ is the multiset of symplectic eigenvalues of $A$. The Inverse Symplectic Eigenvalue Problem of $G$ (ISEP-$G$) is to determine all possible symplectic spectra of a labeled graph $G$ with vertex set $\{1,\dots,2n\}$ (unlike the IEP-$G$, the labeling affects the symplectic spectrum). There are many differences between the IEP-$G$ and the ISEP-$G$, e.g., that there are many labelled graphs that allow all possible symplectic spectra including all symplectic eigenvalues equal. Whereas, for the IEP-$G$, all eigenvalues equal is allowed only by a graph with no edges.
This talk will provide an introduction to the ISEP-$G$, including methods to construct matrices with any given symplectic spectrum and solutions to the ISEP-$G$ for all labeled graphs of order four.