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P. Dueck, S. O'Rourke, D. Renfrew, A. Soshnikov (2011)
Banach Center Publications
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We consider large Wigner random matrices and related ensembles of real symmetric and Hermitian random matrices. Our results are related to the local spectral properties of these ensembles.
Joel A. Tropp (2008)
Studia Mathematica
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This note presents a new proof of an important result due to Bourgain and Tzafriri that provides a partial solution to the Kadison-Singer problem. The result shows that every unit-norm matrix whose entries are relatively small in comparison with its dimension can be paved by a partition of constant size. That is, the coordinates can be partitioned into a constant number of blocks so that the restriction of the matrix to each block of coordinates has norm less than one half. The original...
Martin Ehler, Manuel Gräf, Franz J. Király (2015)
Waves, Wavelets and Fractals
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As a generalization of the standard phase retrieval problem,we seek to reconstruct symmetric rank- 1 matrices from inner products with subclasses of positive semidefinite matrices. For such subclasses, we introduce random cubatures for spaces of multivariate polynomials based on moment conditions. The inner products with samples from sufficiently strong random cubatures allow the reconstruction of symmetric rank- 1 matrices with a decent probability by solving the feasibility problem...
Duheille-Bienvenüe, Frédérique, Guillotin-Plantard, Nadine (2003)
Electronic Communications in Probability [electronic only]
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Oraby, Tamer F. (2007)
Electronic Communications in Probability [electronic only]
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Iványi, Antal (2009)
Acta Universitatis Sapientiae. Mathematica
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L. Pastur (1996)
Annales de l'I.H.P. Physique théorique
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Vladimirov, Igor, Thompson, Bevan (2006)
Journal of Applied Mathematics and Stochastic Analysis
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Soshnikov, Alexander (2004)
Electronic Communications in Probability [electronic only]
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Budden, Mark, Hadavas, Paul, Hoffman, Lorrie (2008)
Applied Mathematics E-Notes [electronic only]
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Brydon Eastman, Kevin N. Vander Meulen (2016)
Special Matrices
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The class of sparse companion matrices was recently characterized in terms of unit Hessenberg matrices. We determine which sparse companion matrices have the lowest bandwidth, that is, we characterize which sparse companion matrices are permutationally similar to a pentadiagonal matrix and describe how to find the permutation involved. In the process, we determine which of the Fiedler companion matrices are permutationally similar to a pentadiagonal matrix. We also describe how to find...
Guntuboyina, Adityanand, Leeb, Hannes (2009)
Electronic Communications in Probability [electronic only]
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Luis Verde-Star (2015)
Special Matrices
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We use elementary triangular matrices to obtain some factorization, multiplication, and inversion properties of triangular matrices. We also obtain explicit expressions for the inverses of strict k-Hessenberg matrices and banded matrices. Our results can be extended to the cases of block triangular and block Hessenberg matrices. An n × n lower triangular matrix is called elementary if it is of the form I + C, where I is the identity matrix and C is lower triangular and has all of its...
Simon Foucart, Ming-Jun Lai (2010)
Studia Mathematica
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For an m × N underdetermined system of linear equations with independent pre-Gaussian random coefficients satisfying simple moment conditions, it is proved that the s-sparse solutions of the system can be found by ℓ₁-minimization under the optimal condition m ≥ csln(eN/s). The main ingredient of the proof is a variation of a classical Restricted Isometry Property, where the inner norm becomes the ℓ₁-norm and the outer norm depends on probability distributions.