科研之友
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摘要
Marcus, Spielman, and Srivastava ['Interlacing families II: mixed characteristic polynomials and the Kadison-Singer problem', Preprint, 2013, arXiv:1306.3969] recently solved the Kadison-Singer problem by showing that if u(1), . . . , u(m) are column vectors in C-d such that Sigma u(i)u(i)(*) = I, then a set of indices S subset of {1, . . . , m} can be chosen so that Sigma(i is an element of S) u(i)u(i)* is approximately 1/2I, with the approximation good in operator norm to order is an element of(1/2) where is an element of = max parallel to u(i) parallel to(2). We extend their result to show that every linear combination of the matrices u(i)u(i)(*) with coefficients in [0, 1] can be approximated in operator norm to order is an element of(1/8) by a matrix of the form Sigma(i is an element of S) u(i)u(i)(*).
- 出版日期2014-6
