T-Jordan Canonical Form and T-Drazin Inverse Based on the T-Product

By Yun Miao, Liqun Qi, Yimin Wei. Communication on Applied Mathematics and Computation, 2019

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In this paper, we investigate the tensor similarity and propose the T-Jordan canonical form and its properties. The concepts of the T-minimal polynomial and the T-characteristic polynomial are proposed. As a special case, we present properties when two tensors commute based on the tensor T-product. We prove that the Cayley–Hamilton theorem also holds for tensor cases. Then, we focus on the tensor decompositions: T-polar, T-LU, T-QR and T-Schur decompositions of tensors are obtained. When an F-square tensor is not invertible with the T-product, we study the T-group inverse and the T-Drazin inverse which can be viewed as the extension of matrix cases. The expressions of the T-group and T-Drazin inverses are given by the T-Jordan canonical form. The polynomial form of the T-Drazin inverse is also proposed. In the last part, we give the T-core-nilpotent decomposition and show that the T-index and T-Drazin inverses can be given by a limit process.

Subfields: Computational Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/33edb995488e42394929082a79b1e8a4a8873583