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Register a new userOn Some Bounds on the Perturbation of Invariant Subspaces of Normal Matrices with Application to a Graph Connection Problem
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Added by
Subhrajit Bhattacharya
Publication date
2021
fields
Informatics Engineering
and research's language is
English
Authors
Subhrajit Bhattacharya
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We provide upper bounds on the perturbation of invariant subspaces of normal matrices measured using a metric on the space of vector subspaces of $mathbb{C}^n$ in terms of the spectrum of both the unperturbed & perturbed matrices, as well as, spectrum of the unperturbed matrix only. The results presented give tighter bounds than the Davis-Khan $sinTheta$ theorem. We apply the result to a graph perturbation problem.
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Given a real-valued positive semidefinite matrix, Williamson proved that it can be diagonalised using symplectic matrices. The corresponding diagonal values are known as the symplectic spectrum. This paper is concerned with the stability of Williamsons decomposition under perturbations. We provide norm bounds for the stability of the symplectic eigenvalues and prove that if $S$ diagonalises a given matrix $M$ to Williamson form, then $S$ is stable if the symplectic spectrum is nondegenerate and $S^TS$ is always stable. Finally, we sketch a few applications of the results in quantum information theory.
Let $A$ be a self-adjoint operator on a Hilbert space $fH$. Assume that the spectrum of $A$ consists of two disjoint components $sigma_0$ and $sigma_1$. Let $V$ be a bounded operator on $fH$, off-diagonal and $J$-self-adjoint with respect to the orthogonal decomposition $fH=fH_0oplusfH_1$ where $fH_0$ and $fH_1$ are the spectral subspaces of $A$ associated with the spectral sets $sigma_0$ and $sigma_1$, respectively. We find (optimal) conditions on $V$ guaranteeing that the perturbed operator $L=A+V$ is similar to a self-adjoint operator. Moreover, we prove a number of (sharp) norm bounds on variation of the spectral subspaces of $A$ under the perturbation $V$. Some of the results obtained are reformulated in terms of the Krein space theory. As an example, the quantum harmonic oscillator under a PT-symmetric perturbation is discussed.
Graph associahedra are generalized permutohedra arising as special cases of nestohedra and hypergraphic polytopes. The graph associahedron of a graph $G$ encodes the combinatorics of search trees on $G$, defined recursively by a root $r$ together with search trees on each of the connected components of $G-r$. In particular, the skeleton of the graph associahedron is the rotation graph of those search trees. We investigate the diameter of graph associahedra as a function of some graph parameters. It is known that the diameter of the associahedra of paths of length $n$, the classical associahedra, is $2n-6$ for a large enough $n$. We give a tight bound of $Theta(m)$ on the diameter of trivially perfect graph associahedra on $m$ edges. We consider the maximum diameter of associahedra of graphs on $n$ vertices and of given tree-depth, treewidth, or pathwidth, and give lower and upper bounds as a function of these parameters. Finally, we prove that the maximum diameter of associahedra of graphs of pathwidth two is $Theta (nlog n)$.
Bounds on the exponential decay of generalized eigenfunctions of bounded and unbounded selfadjoint Jacobi matrices are established. Two cases are considered separately: (i) the case in which the spectral parameter lies in a general gap of the spectrum of the Jacobi matrix and (ii) the case of a lower semi-bounded Jacobi matrix with values of the spectral parameter below the spectrum. It is demonstrated by examples that both results are sharp. We apply these results to obtain a many barriers-type criterion for the existence of square-summable generalized eigenfunctions of an unbounded Jacobi matrix at almost every value of the spectral parameter in suitable open sets. As an application, we provide examples of unbounded Jacobi matrices with a spectral mobility edge.
Necessary and sufficient conditions are given for density of shift-invariant subspaces of the space $mathcal{L}$ of integrable functions of bounded support with the inductive limit topology.
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