We investigate eigenvectors of rank-one deformations of random matrices $\boldsymbol B = \boldsymbol A + \theta \boldsymbol {uu}^{*}$ in which $\boldsymbol A \in \mathbb R^{N \times N}$ is a Wigner real symmetric random matrix, $\theta \in \mathbb R^{+}$ , and $\boldsymbol u$ is uniformly distributed on the unit sphere. It is well known that for $\theta > 1$ the eigenvector associated with the largest eigenvalue of $\boldsymbol B$ closely estimates $\boldsymbol u$ asymptotically, while for $\theta < 1$ the eigenvectors of $\boldsymbol B$ are uninformative about $\boldsymbol u$ . We examine $\mathcal O({1}/{N})$ correlation of eigenvectors with $\boldsymbol u$ before phase transition and show that eigenvectors with larger eigenvalue exhibit stronger alignment with deforming vector through an explicit inverse law ${1}/{\theta ^{*} - x}$ with $\theta ^{*}:= \theta + ({1}/{\theta })$ . This distribution function will be shown to be the ordinary generating function of Chebyshev polynomials of the second kind. These polynomials form an orthogonal set with respect to the semicircle weighting function. This law is an increasing function in the support of semicircle law for eigenvalues $(-2\:,+2)$ . Therefore, most of energy of the unknown deforming vector is concentrated in a $cN$ -dimensional ( $c < 1$ ) known subspace of $\boldsymbol B$ . We use a combinatorial approach to prove the result. We also extend the result to constant rank- $r$ deformations.

中文翻译：

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变形维格纳随机矩阵的特征向量

我们研究随机矩阵的一阶变形的特征向量 $\boldsymbol B = \boldsymbol A + \theta \boldsymbol {uu}^{*}$ 其中 $\boldsymbol A \in \mathbb R^{N \times N}$ 是 Wigner 实对称随机矩阵， $\theta \in \mathbb R^{+}$ ， 和 $\粗体符号 u$ 均匀分布在单位球面上。众所周知，对于 $\theta > 1$ 与最大特征值相关联的特征向量 $\boldsymbol B$ 仔细估计 $\粗体符号 u$ 渐近地，而对于 $\theta < 1$ 的特征向量 $\boldsymbol B$ 不提供信息 $\粗体符号 u$ . 我们检查 $\mathcal O({1}/{N})$ 特征向量与 $\粗体符号 u$ 在相变之前，并通过显式逆定律表明具有较大特征值的特征向量与变形向量表现出更强的对齐 ${1}/{\theta ^{*} - x}$ 和 $\theta ^{*}:= \theta + ({1}/{\theta })$ . 这个分布函数将被证明是第二类切比雪夫多项式的普通生成函数。这些多项式形成关于半圆加权函数的正交集。该定律是支持特征值半圆定律的增函数 $(-2\:,+2)$ . 因此，未知变形矢量的大部分能量集中在一个 $cN$ 维（ $c < 1$ ) 的已知子空间 $\boldsymbol B$ . 我们使用组合方法来证明结果。我们还将结果扩展到恒定等级 - $r$ 变形。