Bounds on the Singular Values of Matrices with Displacement Structure
Bounds on the Singular Values of Matrices with Displacement Structure
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DOI：
10.1137/19m1244433
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发表时间：
2019-06-01
期刊：
影响因子：
10.2
通讯作者：
Townsend, Alex
中科院分区：
文献类型：
--
作者：
Beckermann, Bernhard;Townsend, Alex
具有位移结构的矩阵，例如Pick，Vandermonde和Hankel矩阵，出现在各种应用中。在本文中，我们使用涉及合理功能的极端问题来推导此类矩阵的奇异值的明确界限。例如，我们表明，真实N x N阳性确定的Hankel矩阵H N的KTH奇异值由C rho（-k/log n）平行于H-N与（2）平行，具有明确给定常数C> 0和rho> 1，平行于（2）平行于H-N的是光谱标准。这意味着可以将真实的N x N阳性确定的Hankel矩阵近似，最多可通过等级o（log n log（1/epsilon））矩阵平行于（2）平行于（2）的h-n与（2）平行于（2） 。获得挑选，Cauchy，Real Vandermonde，Lowner和某些Krylov矩阵的类似结果。
Matrices with displacement structure, such as Pick, Vandermonde, and Hankel matrices, appear in a diverse range of applications. In this paper, we use an extremal problem involving rational functions to derive explicit bounds on the singular values of such matrices. For example, we show that the kth singular value of a real n x n positive definite Hankel matrix, H n , is bounded by C rho(-k/log n)parallel to H-n parallel to(2) with explicitly given constants C > 0 and rho > 1, where parallel to H-n parallel to(2) is the spectral norm. This means that a real n x n positive definite Hankel matrix can be approximated, up to an accuracy of epsilon parallel to H-n parallel to(2) with 0 < epsilon < 1, by a rank O(log n log(1/epsilon)) matrix. Analogous results are obtained for Pick, Cauchy, real Vandermonde, Lowner, and certain Krylov matrices.