论文标题
在真正的戴维斯的猜想上
On the real Davies' conjecture
论文作者
论文摘要
我们表明，每个矩阵$ a \ in \ mathbb {r}^{n \ times n} $至少是$δ$δ$ the \ | a \ | $ -close到一个真实矩阵$ a+e \ in \ in \ in \ mathbb {r} $ \ tilde {o} _ {n}（δ^{ - 1}）$。实际上，我们证明，有可能以$ e $的范围成为I.I.D.的足够倍数。界密度的实际下高斯矩阵就足够了。这实质上证实了戴维斯和银行，库尔卡尼，穆克吉和斯里瓦斯塔瓦的猜测，他们最近证明了I.I.D.的结果。复杂的高斯矩阵。 一路上，我们还证明了对随机矩阵的任何两个特征值之间的最小可能距离，其条目具有任意均值；我们论文的这一部分可能具有独立的利益。
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.
