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

Cai, Yunfeng; Cheng, Guanghui; Shi, Decai

doi:10.1002/nla.2238

Cited by 4 publications

(7 citation statements)

References 38 publications

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“…There are several available simultaneous block diagonalization algorithms, see e.g. [79,80,[84][85][86][87][88][89][90][91][92]. An advantage of the Algorithm 3 is that it, with probability 1 and in contrast to Jacobi-like algorithms, is exact; moreover, it can be generalised to find the smallest algebras of observables.…”

Section: A Simultaneous Block Diagonalization and Subalgebras Of Obse...mentioning

confidence: 99%

Constructing k-local parent Lindbladians for matrix product density operators

Bondarenko¹

2021

Preprint

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Section: A Simultaneous Block Diagonalization and Subalgebras Of Obse...mentioning

confidence: 99%

Constructing k-local parent Lindbladians for matrix product density operators

Bondarenko¹

2021

Preprint

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“…Partition Γ = [Γ jk ] with Γ jk ∈ R pj ×p k . Recall (4) and (5), by (P2), we have Γ jk = 0 for j = k, i.e., Γ is τ p -block diagonal; using (P1), Γ = Y diag(γ j Ip j )Y −1 and ∪ j=1 λ(Γ jj ) = λ(Γ), we know that =ˆ , λ(Γ kj kj ) = λ(γ j Ip j ) for 1 ≤ j ≤ , where {k 1 , k 2 , . .…”

Section: Proof Of Theorem 24mentioning

confidence: 99%

“…But this conjecture is only partially proved [29]. Three algebraic methods are proposed to solve bjbdp: When the diagonalizer is orthogonal, using matrix * -algebra, an error controlled method is proposed in [22]; Then the results are non-trivially generalized to the non-orthogonal diagonalizer case in [6]; Using the matrix polynomial, a three-stage method is proposed in [5].…”

Section: A Short Review and Our Contributionmentioning

confidence: 99%

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Perturbation analysis for matrix joint block diagonalization

Cai

2019

Linear Algebra and its Applications

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The matrix joint block diagonalization problem (jbdp) of a given matrix set A = {A i } m i=1 is about finding a nonsingular matrix W such that all W T A i W are block diagonal. It includes the matrix joint diagonalization problem (jdp) as a special case for which all W T A i W are required diagonal. Generically, such a matrix W may not exist, but there are practically applications such as multidimensional independent component analysis (MICA) for which it does exist under the ideal situation, ie., no noise is presented. However, in practice noises do get in and, as a consequence, the matrix set is only approximately block diagonalizable, i.e., one can only make all W T A i W nearly block diagonal at best, where W is an approximation to W , obtained usually by computation. This motivates us to develop a perturbation theory for jbdp to address, among others, the question: how accurate this W is. Previously such a theory for jdp has been discussed, but no effort has been attempted for jbdp yet. In this paper, with the help of a necessary and sufficient condition for solution uniqueness of jbdp recently developed in [Cai and Liu, SIAM J. Matrix Anal. Appl., 38(1):50-71, 2017], we are able to establish an error bound, perform backward error analysis, and propose a condition number for jbdp. Numerical tests validate the theoretical results.

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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%

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

Cited by 4 publications

References 38 publications

Constructing k-local parent Lindbladians for matrix product density operators

Constructing k-local parent Lindbladians for matrix product density operators

Perturbation analysis for matrix joint block diagonalization

Certifying Numerical Decompositions of Compact Group Representations

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