A sum of a large-dimensional random matrix polynomial and a fixed low-rank matrix polynomial is considered. The main assumption is that the resolvent of the random polynomial converges to some deterministic limit. A formula for the limit of the resolvent of the sum is derived, and the eigenvalues are localised. Four instances are considered: a low-rank matrix perturbed by the Wigner matrix, a product HX of a fixed diagonal matrix H and the Wigner matrix X and two special matrix polynomials of higher degree. The results are illustrated with various examples and numerical simulations.

中文翻译：

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矩阵多项式的随机扰动

考虑大维随机矩阵多项式和固定低秩矩阵多项式的和。主要假设是随机多项式的求解器收敛到某个确定性极限。推导了求和解算器的极限公式，并对特征值进行了局部化。考虑了四个实例：一个受 Wigner 矩阵扰动的低秩矩阵、一个固定对角矩阵H和 Wigner 矩阵X的乘积HX以及两个更高次的特殊矩阵多项式。用各种例子和数值模拟来说明结果。