The Expected Norm of Random Matrices
The Expected Norm of Random Matrices
复制标题
随机矩阵的期望范数
DOI：
10.1017/s096354830000420x
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发表时间：
2000
期刊：
影响因子：
--
通讯作者：
Y. Seginer
中科院分区：
文献类型：
--
作者：
Y. Seginer
我们将随机矩阵的欧几里得算子范数与其行和列的欧几里得范数进行比较。在本文的第一部分中，我们证明如果 A 是具有独立同分布的随机矩阵。零均值条目，则 E∥A∥h [les ] Kh (E maxi ∥ai[bull ] ∥h + E maxj ∥aj[bull ] ∥h)，其中 K 是不依赖于 A 的维度或分布的常数（然而，h 确实依赖于维度）。在第二部分中，我们放弃 A 的条目独立同分布的假设。因此，我们考虑通过随机化矩阵条目的符号从（非随机）矩阵获得的随机矩阵 A 的欧几里得算子范数。我们证明，在这种情况下，可能的最佳不等式（直到乘法常数）是 E∥A∥h [les ] (c log1/4 min {m, n})h (E maxi ∥ai[bull ] ∥h + E maxj ∥aj[bull ] ∥h) （m, n 是矩阵的维度，c 是独立于 m, n 的常数）。
We compare the Euclidean operator norm of a random matrix with the Euclidean norm of its rows and columns. In the first part of this paper, we show that if A is a random matrix with i.i.d. zero mean entries, then E∥A∥h [les ] Kh (E maxi ∥ai[bull ] ∥h + E maxj ∥aj[bull ] ∥h), where K is a constant which does not depend on the dimensions or distribution of A (h, however, does depend on the dimensions). In the second part we drop the assumption that the entries of A are i.i.d. We therefore consider the Euclidean operator norm of a random matrix, A, obtained from a (non-random) matrix by randomizing the signs of the matrix's entries. We show that in this case, the best inequality possible (up to a multiplicative constant) is E∥A∥h [les ] (c log1/4 min {m, n})h (E maxi ∥ai[bull ] ∥h + E maxj ∥aj[bull ] ∥h) (m, n the dimensions of the matrix and c a constant independent of m, n).