Perturbation analysis for matrix joint block diagonalization

Cai, Yunfeng; Li, Ren-Cang

doi:10.1016/j.laa.2019.07.012

Cited by 3 publications

(4 citation statements)

References 55 publications

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“…When W is good‐conditioned, ‖ W −1 X ‖ F is small. However, if W is ill‐conditioned, ‖ W −1 X ‖ F can be quite large, which means that X Π can be of “low quality.” In fact, from the perturbation theory of the JBD problem (see also the perturbation theory of the JD problem in the works of Shi et al and Afsari), the diagonalizer is sensitive to the perturbation when it is ill‐conditioned. Therefore, it is not surprising to conclude that C can be large when W is ill‐conditioned.…”

Section: Resultsmentioning

confidence: 99%

Solving the general joint block diagonalization problem via linearly independent eigenvectors of a matrix polynomial

Cai

Cheng

Shi

2019

Numerical Linear Algebra App

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Summary In this paper, we consider the exact/approximate general joint block diagonalization (GJBD) problem of a matrix set false{Aifalse}i=0p ( p ≥ 1), where a nonsingular matrix W (often referred to as a diagonalizer) needs to be found such that the matrices W HAiW 's are all exactly/approximately block‐diagonal matrices with as many diagonal blocks as possible. We show that the diagonalizer of the exact GJBD problem can be given by W = [x1,x2,…,xn]Π, where Π is a permutation matrix and xi's are eigenvectors of the matrix polynomial Pfalse(λfalse)=∑i=0pλiAi, satisfying that [x1,x2,…,xn] is nonsingular and where the geometric multiplicity of each λi corresponding with xi is equal to 1. In addition, the equivalence of all solutions to the exact GJBD problem is established. Moreover, a theoretical proof is given to show why the approximate GJBD problem can be solved similarly to the exact GJBD problem. Based on the theoretical results, a three‐stage method is proposed, and numerical results show the merits of the method.

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Section: Resultsmentioning

confidence: 99%

Solving the general joint block diagonalization problem via linearly independent eigenvectors of a matrix polynomial

Cai

Cheng

Shi

2019

Numerical Linear Algebra App

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“…This requires finding a common block-diagonalization of the matrices representing the symmetry group action. A large number of numerical methods for this task have been developed [10][11][12][13][14][15][16][17][18][19][20][21][22][23]. These algorithms must be compared along a number of different dimensions:…”

Section: Introductionmentioning

confidence: 99%

“…References [20][21][22][23] give algorithms for finding a block decomposition for general * -algebras and come with rigorous guarantees. Refs.…”

Section: Introductionmentioning

confidence: 99%

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Certifying Numerical Decompositions of Compact Group Representations

Montealegre-Mora,

Rosset,

Bancal

et al. 2021

Preprint

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We present a performant and rigorous algorithm for certifying that a matrix is close to being a projection onto an irreducible subspace of a given group representation. This addresses a problem arising when one seeks solutions to semi-definite programs (SDPs) with a group symmetry. Indeed, in this context, the dimension of the SDP can be significantly reduced if the irreducible representations of the group action are explicitly known. Rigorous numerical algorithms for decomposing a given group representation into irreps are known, but fairly expensive. To avoid this performance problem, existing software packages -e.g. RepLAB, which motivated the present work -use randomized heuristics. While these seem to work well in practice, the problem of to which extent the results can be trusted arises. Here, we provide rigorous guarantees applicable to finite and compact groups, as well as a software implementation that can interface with RepLAB. Under natural assumptions, a commonly used previous method due to Babai and Friedl runs in time O(n 5 ) for n-dimensional representations. In our approach, the complexity of running both the heuristic decomposition and the certification step is O(max{n 3 log n, D d 2 log d}), where d is the maximum dimension of an irreducible subrepresentation, and D is the time required to multiply elements of the group. A reference implementation interfacing with RepLAB is provided.

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The Environmental Hazards and Treatment of Ship’s Domestic Sewage

Zhang,

Xian,

Sun

et al. 2024

Toxics

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With the rapid development of the modern shipping field, the damage caused by ship pollution to the global inland waterways and marine ecosystems has attracted extensive attention from the international community. However, there are fewer reviews on the environmental hazards of domestic ship sewage and its treatment, and a systematic summary of the environmental hazards posed by ship domestic sewage and its treatment is lacking. Based on summarizing the various environmental hazards brought about by a ship’s domestic sewage and the corresponding treatment methods, this study elaborates, in detail, on the specific hazards of the main toxic and hazardous substances contained in a ship’s domestic sewage on the environment and organisms, and the treatment methods of the ship’s domestic sewage and their treatment effects, such as membrane bioreactor (MBR). It is also pointed out that MBR has great potential in the direction of ship domestic sewage treatment, and the solution of its membrane pollution and other problems as well as the exploration of the combination of MBR and other treatment methods will become the focus of future research. A theoretical reference is provided for the study of environmental problems caused by domestic sewage from ships and their treatment options.

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Perturbation analysis for matrix joint block diagonalization

Cited by 3 publications

References 55 publications

Solving the general joint block diagonalization problem via linearly independent eigenvectors of a matrix polynomial

Solving the general joint block diagonalization problem via linearly independent eigenvectors of a matrix polynomial

Certifying Numerical Decompositions of Compact Group Representations

The Environmental Hazards and Treatment of Ship’s Domestic Sewage

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