Improving upon results of Rudelson and Vershynin, we establish delocalization bounds for eigenvectors of independent‐entry random matrices. In particular, we show that with high probability every eigenvector is delocalized, meaning any subset of its coordinates carries an appropriate proportion of its mass. Our results hold for random matrices with genuinely complex as well as real entries. As an application of our methods, we also establish delocalization bounds for normal vectors to random hyperplanes. The proofs of our main results rely on a least singular value bound for genuinely complex rectangular random matrices, which generalizes a previous bound due to the first author, and may be of independent interest.

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非Hermitian随机矩阵的特征向量离域及其应用

根据Rudelson和Vershynin的结果，我们为独立输入随机矩阵的特征向量建立了离域边界。尤其是，我们表明，每个特征向量都有很高的离域性，这意味着其坐标的任何子集都具有适当比例的质量。我们的结果适用于具有真正复杂项和真实项的随机矩阵。作为我们方法的应用，我们还建立了法向向量到随机超平面的离域边界。我们主要结果的证明依赖于真正复杂的矩形随机矩阵的最小奇异值边界，该矩阵泛化了归因于第一作者的先前边界，并且可能具有独立利益。