关于实Davies猜想
On the real Davies' conjecture
摘要：
我们证明了每个矩阵$A \in \mathbb{R}^{n\times n}$至少是$δ$ $\|A\|$ -接近一个实矩阵$A+E \in \mathbb{R}^{n\times n}$，其特征向量的条件数至多为$\tilde{O}_{n}(δ^{-1})$。事实上，我们证明，以高概率，采取$E$是一个足够小的倍数的i.i.d.有界密度的实次高斯矩阵就足够了。这基本上证实了Davies以及Banks、Kulkarni、Mukherjee和Srivastava的推测，他们最近证明了i.i.d.的这一结果。复高斯矩阵一路上，我们还证明了非渐近估计的最小可能的距离之间的任意两个特征值的随机矩阵的条目有任意的手段，我们的论文的这一部分可能是独立的兴趣。
英文摘要：
We show that every matrix $A \in \mathbb{R}^{n\times n}$ is at least $δ$$\|A\|$-close to a real matrix $A+E \in \mathbb{R}^{n\times n}$ whose eigenvectors have condition number at most $\tilde{O}_{n}(δ^{-1})$. In fact, we prove that, with high probability, taking $E$ to be a sufficiently small multiple of an i.i.d. real sub-Gaussian matrix of bounded density suffices. This essentially confirms a speculation of Davies, and of Banks, Kulkarni, Mukherjee, and Srivastava, who recently proved such a result for i.i.d. complex Gaussian matrices. Along the way, we also prove non-asymptotic estimates on the minimum possible distance between any two eigenvalues of a random matrix whose entries have arbitrary means; this part of our paper may be of independent interest.
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