Optical properties of materials related to light absorption and scattering are explained by the excitation of electrons. The Bethe–Salpeter equation is the state-of-the-art approach to describe these processes from first principles (ab initio), i.e. without the need for empirical data in the model. To harness the predictive power of the equation, it is mapped to an eigenvalue problem via an appropriate discretization scheme. The eigenpairs of the resulting large, dense, structured matrix can be used to compute dielectric properties of the considered crystalline or molecular system. The matrix always shows a 2 × 2 block structure. Depending on exact circumstances and discretization schemes, one ends up with a matrix structure such as $H_1=\left[\begin{array}{cc}\hfill A\hfill & \hfill B\hfill \\ \hfill -B\hfill & \hfill -A\hfill \end{array}\right]\in {ℂ}^{2n\times 2n},A=A^{\mathsf{H}},B=B^{\mathsf{H}},$$\text{or}{H}_2=\left[\begin{array}{cc}\hfill A\hfill & \hfill B\hfill \\ \hfill -B^{\mathsf{H}}\hfill & \hfill -A^{\mathsf{T}}\hfill \end{array}\right]\in {ℂ}^{2n\times 2n}\text{or}{\mathbb{R}}^{2n\times 2n},A=A^{\mathsf{H}},B=B^{\mathsf{T}}.$ $H_1$ can be acquired for crystalline systems (see Sander et al. (2015)), $H_2$ is a more general form found e.g. in Shao et al. (2016) and Penke et al. (2020), which can for example be used to study molecules. Additionally, certain definiteness properties may hold. In this work, we compile theoretical results characterizing the structure of $H_1$ and $H_2$ in the language of non-standard scalar products. These results enable us to develop a generalized perspective on the currently used direct solution approach for matrices of form $H_1$. This new viewpoint is used to develop two alternative methods for solving the eigenvalue problem. Both have advantages over the method currently in use and are well suited for high performance environments and only rely on basic numerical linear algebra building blocks. The results are extended to hold even without the mentioned definiteness property, showing the usefulness of our new perspective.

与光吸收和散射相关的材料的光学特性可以通过电子的激发来解释。Bethe-Salpeter 方程是从第一性原理 ( ab initio )描述这些过程的最新方法，即不需要模型中的经验数据。为了利用方程的预测能力，通过适当的离散化方案将其映射到特征值问题。生成的大、致密、结构化矩阵的特征对可用于计算所考虑的晶体或分子系统的介电特性。矩阵始终显示 2 × 2 块结构。根据具体情况和离散化方案，最终会得到一种矩阵结构，例如$H_1=\left[\begin{array}{cc}\hfill \mathrm{一种}\hfill & \hfill 乙\hfill \\ \hfill -乙\hfill & \hfill -\mathrm{一种}\hfill \end{array}\right]\in {ℂ}^{2n\times 2n},\mathrm{一种}={\mathrm{一种}}^{\mathsf{H}},乙={乙}^{\mathsf{H}},$$\text{要么}{H}_2=\left[\begin{array}{cc}\hfill \mathrm{一种}\hfill & \hfill 乙\hfill \\ \hfill -{乙}^{\mathsf{H}}\hfill & \hfill -{\mathrm{一种}}^{\mathsf{吨}}\hfill \end{array}\right]\in {ℂ}^{2n\times 2n}\text{要么}{\mathbb{电阻}}^{2n\times 2n},\mathrm{一种}={\mathrm{一种}}^{\mathsf{H}},乙={乙}^{\mathsf{吨}}.$ $H_1$ 可以为结晶系统获得（参见 Sander 等人（2015 年））， $H_2$是一种更一般的形式，例如在 Shao 等人中发现的。(2016) 和 Penke 等人。（2020），例如可用于研究分子。此外，某些确定性属性可能成立。在这项工作中，我们汇编了表征结构的理论结果$H_1$ 和 $H_2$在非标准标量产品的语言中。这些结果使我们能够对当前使用的形式矩阵的直接求解方法形成一个广义的观点$H_1$. 这个新观点被用来开发解决特征值问题的两种替代方法。两者都优于当前使用的方法，并且非常适合高性能环境，并且仅依赖于基本的数值线性代数构建块。即使没有提到的确定性，结果也被扩展为成立，显示了我们新视角的有用性。