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

Yun Miao; Liqun Qi; Yimin Wei

doi:10.1007/s42967-019-00055-4

Communications on Applied Mathematics and Computation ›› 2020, Vol. 3 ›› Issue (2) : 201 -220. DOI: 10.1007/s42967-019-00055-4

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

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Abstract

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.

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Yun Miao, Liqun Qi, Yimin Wei. T-Jordan Canonical Form and T-Drazin Inverse Based on the T-Product. Communications on Applied Mathematics and Computation, 2020, 3(2): 201-220 DOI:10.1007/s42967-019-00055-4