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：本文研究了多线性代数在多维线性时不变（MLTI）系统降阶中的应用。我们以张量Krylov子空间方法为核心工具，通过低维子空间逼近系统解。引入张量扩展块Krylov子空间与全局Krylov子空间，以及相应的基于Arnoldi的算法流程。利用这些方法，我们构建了基于投影技术的模型降阶方案，并阐明了如何将其用于求解平衡截断法（一种典型降阶技术）所需的大规模Lyapunov张量方程。通过爱因斯坦积，我们演示了如何利用张量扩展块Arnoldi算法与扩展全局Arnoldi算法提取近似解。