The random paving property for uniformly bounded matrices

Tom 185 / 2008

Joel A. Tropp Studia Mathematica 185 (2008), 67-82 MSC: 46B09, 15A52. DOI: 10.4064/sm185-1-4


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 proof of Bourgain and Tzafriri involves a long, delicate calculation. The new proof relies on the systematic use of symmetrization and (noncommutative) Khinchin inequalities to estimate the norms of some random matrices.


  • Joel A. TroppApplied & Computational Mathematics, MC 217-50
    California Institute of Technology
    1200 E. California Blvd.
    Pasadena, CA 91125-5000, U.S.A.

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