In this work, we explore the application of multilinear algebra in reducing the order of multidimentional linear time-invariant (MLTI) systems. We use tensor Krylov subspace methods as key tools, which involve approximating the system solution within a low-dimensional subspace. We introduce the tensor extended block and global Krylov subspaces and the corresponding Arnoldi based processes. Using these methods, we develop a model reduction using projection techniques. We also show how these methods could be used to solve large-scale Lyapunov tensor equations that are needed in the balanced truncation method which is a technique for order reduction. We demonstrate how to extract approximate solutions via the Einstein product using the tensor extended block Arnoldi and the extended global Arnoldi processes.

翻译：本文探讨了多线性代数在多维线性时不变系统降阶中的应用。我们以张量Krylov子空间方法为核心工具，其基本思想是在低维子空间内对系统解进行逼近。引入张量扩展分块Krylov子空间和张量扩展全局Krylov子空间，并给出相应的基于Arnoldi过程的实现方法。基于这些方法，我们发展了一种基于投影技术的模型降阶方案。同时，展示了如何将这些方法用于求解平衡截断法（一种降阶技术）所需的大规模Lyapunov张量方程。最后，通过张量扩展分块Arnoldi过程和扩展全局Arnoldi过程，演示了如何利用Einstein积提取近似解。