Almost orthogonal submatrices of an orthogonal matrix
Almost orthogonal submatrices of an orthogonal matrix
复制标题
正交矩阵的几乎正交子矩阵
DOI：
10.1007/bf02810682
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发表时间：
1996
影响因子：
1
通讯作者：
M. Rudelson
中科院分区：
文献类型：
--
作者：
M. Rudelson
摘要Lett≥1且设n、M为自然数，n<M。 Leta=(ai,j) 是一个 n xM 矩阵，其行是正交的。假设 A 列的 ℓ2-范数是一致有界的。即，对于 allj
$$\sqrt {\frac{M}{n}} \cdot \left( {\sum\limits_{i = 1}^n {a_{i,j}^2 } } \right)^{1/2} \leqslant t.$$
使用主要测度估计，我们证明对于每个 ε>0，最多存在一组基数 I ⊃ {1,…,M}
$$C \cdot \frac{{t^2 }}{{\varepsilon ^2 }} \cdot n \cdot log\frac{{nt^2 }}{{\varepsilon ^2 }}$$
使得矩阵
$$\sqrt {M/\left| I \right|} \cdot A_I^T $$
，其中AI=(ai,j)j∈I，充当从 ℓ2n 到 (1+ε)-同构
$$\ell _2^{\反斜杠 I\反斜杠 } $$
。
AbstractLett≥1 and letn, M be natural numbers,n<M. Leta=(ai,j) be ann xM matrix whose rows are orthonormal. Suppose that the ℓ2-norms of the columns ofA are uniformly bounded. Namely, for allj
$$\sqrt {\frac{M}{n}} \cdot \left( {\sum\limits_{i = 1}^n {a_{i,j}^2 } } \right)^{1/2} \leqslant t.$$
Using majorizing measure estimates we prove that for every ε>0 there exists, a setI ⊃ {1,…,M} of cardinality at most
$$C \cdot \frac{{t^2 }}{{\varepsilon ^2 }} \cdot n \cdot log\frac{{nt^2 }}{{\varepsilon ^2 }}$$
such that the matrix
$$\sqrt {M/\left| I \right|} \cdot A_I^T $$
, whereAI=(ai,j)j∈I, acts as a (1+ε)-isomorphism from ℓ2n into
$$\ell _2^{\backslash I\backslash } $$
.