# A New Approach to Multilinear Dynamical Systems and Control

arXiv: https://arxiv.org/abs/2108.13583 · HTML: https://mlsanity.com/paper/2108.13583/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 2 of its 32 references that are in the map.
- First posted on arXiv 31 Aug 2021.

## Builds on (most cited first, 2 of 2)

- [Transform-Based Multilinear Dynamical System for Tensor Time Series Analysis](https://mlsanity.com/paper/1811.07342/) (2018)
- [Tensor Decompositions for Modeling Inverse Dynamics](https://mlsanity.com/paper/1711.04683/) (2017)

## Built on by (most cited first, 1 of 1)

- [Causal Deep Learning](https://mlsanity.com/paper/2301.00314/) (2023)

## Abstract (arXiv)

The current paper presents a new approach to multilinear dynamical systems analysis and control. The approach is based upon recent developments in tensor decompositions and a newly defined algebra of circulants. In particular, it is shown that under the right tensor multiplication operator, a third order tensor can be written as a product of third order tensors that is analogous to a traditional matrix eigenvalue decomposition where the "eigenvectors" become eigenmatrices and the "eigenvalues" become eigen-tuples. This new development allows for a proper tensor eigenvalue decomposition to be defined and has natural extension to linear systems theory through a textittensor-exponential. Through this framework we extend many of traditional techniques used in linear system theory to their multilinear counterpart.