We are concerned with the eigenvector-dependent nonlinear eigenvalue problem (NEPv) $H\left(V\right)V=V\Lambda$, where $H\left(V\right)\in {ℂ}^{n\times n}$ is a Hermitian matrix-valued function of $V\in {ℂ}^{n\times k}$ with orthonormal columns, i.e., $V^{H}V=I_k$, $k\le n$ (usually $k\ll n$). Sufficient conditions on the solvability and solution uniqueness of NEPv are obtained, based on the well-known results from the relative perturbation theory. These results are complementary to recent ones in Cai et al. (2018), where, among others, one can find conditions for the solvability and solution uniqueness of NEPv, based on the well-known results from the absolute perturbation theory. Although the absolute perturbation theory is more versatile in applications, there are cases where the relative perturbation theory produces better results.

我们关注与特征向量有关的非线性特征值问题（NEPv） $H（\mathrm{伏特}）\mathrm{伏特}=\mathrm{伏特}\Lambda$， 在哪里 $H（\mathrm{伏特}）\in {ℂ}^{ñ\times ñ}$ 是Hermitian矩阵值的函数 $\mathrm{伏特}\in {ℂ}^{ñ\times ķ}$ 具有正交列，即 ${\mathrm{伏特}}^H\mathrm{伏特}={\mathrm{一世}}_{ķ}$， $ķ\le ñ$ （通常 $ķ\ll ñ$）。基于相对摄动理论的众所周知的结果，获得了关于NEPv的可溶性和溶液唯一性的充分条件。这些结果与Cai等人的最新研究结果相辅相成。（2018），其中，可以基于绝对扰动理论的众所周知的结果找到NEPv的可溶性和溶液唯一性的条件。尽管绝对扰动理论在应用中用途更广，但在某些情况下，相对扰动理论会产生更好的结果。