Matrix Bruhat decompositions with a remark on the QR (GR) algorithm
TLDR
It is shown that it is the modified Bruhat decomposition that governs the eigenvalue disorder in the QR (GR) algorithm.About:
This article is published in Linear Algebra and its Applications.The article was published on 1997-01-01 and is currently open access. It has received 13 citations till now. The article focuses on the topics: QR algorithm & QR decomposition.read more
Citations
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Geometric Structure of High-Dimensional Data and Dimensionality Reduction
TL;DR: The book moreover stresses the recently developed nonlinear methods and introduces the applications of dimensionality reduction in many areas, such as face recognition, image segmentation, data classification, data visualization, and hyperspectral imagery data analysis.
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Fast computation of the rank profile matrix and the generalized Bruhat decomposition
TL;DR: It is shown how a PLUQ decomposition revealing the rank profile matrix also reveals both a row and a column echelon form of the input matrix or of any of its leading sub-matrices, by a simple post-processing made of row and column permutations.
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Time and space efficient generators for quasiseparable matrices
Clément Pernet,Arne Storjohann +1 more
TL;DR: In this paper, the rank profile matrix invariant (RPMI) is introduced and two new structured representations for exact linear algebra are presented. But the connection between the notion of quasiseparability and the rank-profile matrix invariance is not discussed.
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Bruhat canonical form for linear systems
W. Manthey,Uwe Helmke +1 more
TL;DR: In this paper, a new canonical form for state space equivalence of controllable and observable linear systems is introduced, which is closely related to a canonical form due to Bosgra and van der Weiden.
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Systematic maximum sum rank codes
TL;DR: In this article, the algebraic properties and representation of encoders in systematic form of maximum rank distance (MRD) codes and Maximum Sum Rank Distance (MSRD) codes are investigated.
References
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The algebraic eigenvalue problem
TL;DR: Theoretical background Perturbation theory Error analysis Solution of linear algebraic equations Hermitian matrices Reduction of a general matrix to condensed form Eigenvalues of matrices of condensed forms The LR and QR algorithms Iterative methods Bibliography.
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The Symmetric Eigenvalue Problem.
TL;DR: Parlett as discussed by the authors presents mathematical knowledge that is needed in order to understand the art of computing eigenvalues of real symmetric matrices, either all of them or only a few.
Book
The Symmetric Eigenvalue Problem
TL;DR: Parlett as discussed by the authors presents mathematical knowledge that is needed in order to understand the art of computing eigenvalues of real symmetric matrices, either all of them or only a few.