Quasi-cyclic displacement and inversion decomposition of a quasi-Toeplitz matrix

Yanpeng Zheng; Xiaoyu Jiang; Yanpeng Zheng; Xiaoyu Jiang

doi:10.3934/math.2022649

AIMS Mathematics

2022, Volume 7, Issue 7: 11647-11662. doi: 10.3934/math.2022649

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Research article

Quasi-cyclic displacement and inversion decomposition of a quasi-Toeplitz matrix

Yanpeng Zheng ¹,
Xiaoyu Jiang ^{2
,
,}

1.
School of Automation and Electrical Engineering Linyi University, Linyi, 276000, China
2.
School of Information Science and Engineering Linyi University, Linyi, 276000, China

Received: 28 December 2021 Revised: 21 March 2022 Accepted: 05 April 2022 Published: 15 April 2022
MSC : 47C05, 47C15, 15A09, 15B05

We study a class of column upper-minus-lower (CUML) Toeplitz matrices, which are "close" to the Toeplitz matrices in the sense that their ($ 1, -1 $)-cyclic displacements coincide with $ \varphi $-cyclic displacement of some Toeplitz matrices. Among others, we derive the inverse formula for CUML Toeplitz matrices in the form of sums of products of factor circulants by constructing the corresponding displacement of the matrices. In addition, by the relationship between CUML Toeplitz matrices and CUML Hankel matrices, the inverse formula for CUML Hankel matrices is also obtained.
- CUML Toeplitz matrix,
- CUML Hankel matrix,
- RFMLR circulants,
- RSFPLR circulants,
- Cyclic displacement,
- inverse
Citation: Yanpeng Zheng, Xiaoyu Jiang. Quasi-cyclic displacement and inversion decomposition of a quasi-Toeplitz matrix[J]. AIMS Mathematics, 2022, 7(7): 11647-11662. doi: 10.3934/math.2022649

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Abstract

We study a class of column upper-minus-lower (CUML) Toeplitz matrices, which are "close" to the Toeplitz matrices in the sense that their ($ 1, -1 $)-cyclic displacements coincide with $ \varphi $-cyclic displacement of some Toeplitz matrices. Among others, we derive the inverse formula for CUML Toeplitz matrices in the form of sums of products of factor circulants by constructing the corresponding displacement of the matrices. In addition, by the relationship between CUML Toeplitz matrices and CUML Hankel matrices, the inverse formula for CUML Hankel matrices is also obtained.

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