Communications in Mathematical Sciences

Volume 20 (2022)

Number 3

ST-SVD factorization and s-diagonal tensors

Pages: 597 – 610



Chen Ling (Department of Mathematics, Hangzhou Dianzi University, Hangzhou, China)

Jinjie Liu (School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai, China)

Chen Ouyang (School of Computer Science and Technology, Dongguan University of Technology, Dongguan, China)

Liqun Qi (Department of Mathematics, Hangzhou Dianzi University, Hangzhou, China; and Department of Applied Mathematics, Hong Kong Polytechnic University, Kowloon, Hong Kong)


A third order real tensor is mapped to a special f-diagonal tensor by going through Discrete Fourier Transform (DFT), standard matrix SVD and inverse DFT. We call such an f‑diagonal tensor an s‑diagonal tensor. An f‑diagonal tensor is an s‑diagonal tensor if and only if it is mapped to itself in the above process. The third order tensor space is partitioned into orthogonal equivalence classes. Each orthogonal equivalence class has a unique s‑diagonal tensor. Two s‑diagonal tensors are equal if they are orthogonally equivalent. Third order tensors in an orthogonal equivalence class have the same tensor tubal rank and T‑singular values. Four meaningful necessary conditions for s‑diagonal tensors are presented. Then we present a set of sufficient and necessary conditions for s‑diagonal tensors. Such conditions involve a special complex number. In the cases that the dimension of the third mode of the considered tensor is $2$, $3$ and $4$, we present direct sufficient and necessary conditions which do not involve such a complex number.


T-SVD factorization, s-diagonal tensor, f-diagonal tensor, necessary conditions, sufficient and necessary conditions

2010 Mathematics Subject Classification

15A18, 15A69

Ling Chen’s work was supported by Natural Science Foundation of China (No. 11971138) and Natural Science Foundation of Zhejiang Province (Nos. LY19A010019, LD19A010002).

Jinjie Liu’s work was supported by Natural Science Foundation of China (No. 12001366).

Chen Ouyang’s work was supported by Natural Science Foundation of China (No. 11971106).

Received 17 April 2021

Received revised 4 August 2021

Accepted 8 August 2021

Published 21 March 2022