Computation on GPU of Eigenvalues and Eigenvectors of a Large Number of Small Hermitian Matrices

Abstract The classical Aronszajn–Donoghue theorem states that for a rank-one perturbation of a self-adjoint operator (by a cyclic vector) the singular parts of the spectral measures of the original and perturbed operators are mutually singular. As simple direct sum type examples show, this result does not hold for finite rank perturbations. However, the set of exceptional perturbations is pretty small. Namely, for a family of rank $d$ perturbations $A_{\boldsymbol{\alpha }}:= A + {\textbf{B}} {\boldsymbol{\alpha }} {\textbf{B}}^*$, ${\textbf{B}}:{\mathbb C}^d\to{{\mathcal{H}}}$, with ${\operatorname{Ran}}{\textbf{B}}$ being cyclic for $A$, parametrized by $d\times d$ Hermitian matrices ${\boldsymbol{\alpha }}$, the singular parts of the spectral measures of $A$ and $A_{\boldsymbol{\alpha }}$ are mutually singular for all ${\boldsymbol{\alpha }}$ except for a small exceptional set $E$. It was shown earlier by the 1st two authors, see [4], that $E$ is a subset of measure zero of the space $\textbf{H}(d)$ of $d\times d$ Hermitian matrices. In this paper, we show that the set $E$ has small Hausdorff dimension, $\dim E \le \dim \textbf{H}(d)-1 = d^2-1$.

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Matching Point Clouds with STL Models by Using the Principle Component Analysis and a Decomposition into Geometric Primitives

Applied Sciences ◽

10.3390/app11052268 ◽

2021 ◽

Vol 11 (5) ◽

pp. 2268

Author(s):

Erika Straková ◽

Dalibor Lukáš ◽

Zdenko Bobovský ◽

Tomáš Kot ◽

Milan Mihola ◽

...

Keyword(s):

Point Cloud ◽

Principal Component ◽

Point Clouds ◽

Component Analysis ◽

Language Models ◽

Computational Time ◽

Dominant Eigenvalue ◽

Eigenvalues And Eigenvectors ◽

Correct Model ◽

Geometric Primitives

While repairing industrial machines or vehicles, recognition of components is a critical and time-consuming task for a human. In this paper, we propose to automatize this task. We start with a Principal Component Analysis (PCA), which fits the scanned point cloud with an ellipsoid by computing the eigenvalues and eigenvectors of a 3-by-3 covariant matrix. In case there is a dominant eigenvalue, the point cloud is decomposed into two clusters to which the PCA is applied recursively. In case the matching is not unique, we continue to distinguish among several candidates. We decompose the point cloud into planar and cylindrical primitives and assign mutual features such as distance or angle to them. Finally, we refine the matching by comparing the matrices of mutual features of the primitives. This is a more computationally demanding but very robust method. We demonstrate the efficiency and robustness of the proposed methodology on a collection of 29 real scans and a database of 389 STL (Standard Triangle Language) models. As many as 27 scans are uniquely matched to their counterparts from the database, while in the remaining two cases, there is only one additional candidate besides the correct model. The overall computational time is about 10 min in MATLAB.

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Eigenbranes in Jackiw-Teitelboim gravity

Journal of High Energy Physics ◽

10.1007/jhep02(2021)168 ◽

2021 ◽

Vol 2021 (2) ◽

Author(s):

Andreas Blommaert ◽

Thomas G. Mertens ◽

Henri Verschelde

Keyword(s):

Finite Volume ◽

Chaotic System ◽

Path Integral ◽

Hermitian Matrices ◽

The Matrix ◽

Integral Contour

Abstract It was proven recently that JT gravity can be defined as an ensemble of L × L Hermitian matrices. We point out that the eigenvalues of the matrix correspond in JT gravity to FZZT-type boundaries on which spacetimes can end. We then investigate an ensemble of matrices with 1 ≪ N ≪ L eigenvalues held fixed. This corresponds to a version of JT gravity which includes N FZZT type boundaries in the path integral contour and which is found to emulate a discrete quantum chaotic system. In particular this version of JT gravity can capture the behavior of finite-volume holographic correlators at late times, including erratic oscillations.

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