Building on previous work that provided analytical solutions to generalised matrix eigenvalue problems arising from numerical discretisations, this paper develops exact eigenvalues and eigenvectors for a broader class of $n$-dimensional matrices, focusing on non-symmetric and non-persymmetric matrices. These matrices arise in one-dimensional Laplacian eigenvalue problems with mixed boundary conditions and in a few quantum mechanics applications where standard Toeplitz-plus-Hankel matrix forms do not suffice. By extending analytical methodologies to these broader matrix categories, the study not only widens the scope of applicable matrices but also enhances computational methodologies, leading to potentially more accurate and efficient solutions in physics and engineering simulations.

翻译：基于先前为数值离散化产生的广义矩阵特征值问题提供解析解的研究，本文针对更广泛的一类$n$维矩阵发展了精确的特征值与特征向量，主要关注非对称与非逆对称矩阵。这类矩阵出现在具有混合边界条件的一维拉普拉斯特征值问题中，以及一些标准Toeplitz加Hankel矩阵形式不满足需求的量子力学应用中。通过将解析方法扩展至这些更广泛的矩阵类别，本研究不仅拓宽了适用矩阵的范围，还增强了计算方法学，有望为物理学与工程模拟提供更精确高效的解决方案。