2019/02/19 by Yun Miao, Liqun Qi, Miao, Yun +3 · 1 citation
Computer Science · Mathematics · #15A48 #15A69 #65F10 #65H10 #65N22 #FOS: Mathematics #Matrix Theory and Algorithms #Numerical Analysis (math.NA) #Tensor decomposition and applications
paper · pdf · doi:10.48550/arxiv.1902.07024
openalex publication_date 2019/02/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we introduce the tensor similar transforation and propose the T-Jordan canonical form and its properties. The concept of T-minimal polynomial and T-characteristic polynomial are raised. As a special case, we present properties when two tensors commutes via the tensor T-product. The Cayley-Hamilton theorem also holds for tensor cases. Then we focus on the tensor decomposition theory. T-polar, T-LU, T-QR and T-Schur decomposition of tensors are obtained. When a F-square tensor is not invertible via the T-product, we give the T-group inverse and T-Drazin inverse which can be viewed as the extension of matrix cases. The expression of T-group and T-Drazin inverse are given by the T-Jordan canonical form. The polynomial form of T-Drazin inverse is also proposed. As an application we raise the T-linear system and get its solution. In the last part, we give the T-core-nilpotent decomposition and show that the T-index and T-Drazin inverse can be given by a limitation process.