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.

Keywords:

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

Since its initiation in 1979, the Canadian Applied and Industrial Mathematics Society — Société Canadienne de Mathématiques Appliquées et Industrielles (CAIMS–SCMAI) has gained a growing presence in industrial, mathematical, scientific, and technological circles within and outside of Canada. Its members contribute to state-of-the-art research in industry, natural sciences, medicine and health, finance, physics, engineering, and more. The annual meetings are a highlight of the year. CAIMS–SCMAI is an active member society of the International Council for Industrial and Applied Mathematics, which hosts the prestigious ICIAM Congresses every four years.

Canadian Applied and Industrial Mathematics is at the forefront of scientific and technological development. We use advanced mathematics to tackle real-world problems in science and industry and develop new theories to analyse structures that arise from the modelling of real-world problems.

Applied Mathematics has evolved from traditional applications in areas such as fluids, mechanics, and physics, to modern topics such as medicine, health, biology, data science, finance, nano-tech, etc. Its growing importance in all aspects of life, health, and management increases the need for publication venues for high-level applied and industrial mathematics. Hence CAIMS–SCMAI decided to start a scientific journal called Mathematics in Science and Industry (MSI) to add value to the discussion of applied and industrial mathematics worldwide.

Submissions to MSI in all areas of applied and industrial mathematics are welcome (https://caims.ca/mathematics_in_science_and_industry/). We offer a timely and high-quality review process, and papers are published online as open access, with the publication fee being covered by CAIMS for the first five years.

MSI is honored that leading experts in industrial and applied mathematics have offered their support as editors:

Editors in Chief:

● Thomas Hillen (University of Alberta, thillen@ualberta.ca)

● Ray Spiteri (University of Saskatchewan, spiteri@cs.usask.ca)

Associate Editors:

● Lia Bronsard (McMaster University)

● Richard Craster (Imperial College of London, UK)

● David Earn (McMaster University)

● Ronald Haynes (Memorial University)

● Jane Heffernan (York University)

● Nicholas Kevlahan (McMaster University)

● Yong-Jung Kim (KAIST, Korea)

● Mark Lewis (University of Alberta)

● Kevin J. Painter (Heriot-Watt University, UK)

● Vakhtang Putkaradze (ATCO)

● Katrin Rohlf (Ryerson University)

● John Stockie (Simon Fraser University)

● Jie Sun (Huawei, Hong Kong)

● Justin Wan (University of Waterloo)

● Michael Ward (University of British Columbia)

● Tony Ware (University of Calgary)

● Brian Wetton (University of British Columbia)

The first eight papers of MSI, presented here, are published as special issue in AIMS Mathematics. They showcase a broad representation of applied mathematics that touches the interests of Canadian researchers and our many collaborators around the world. The science that we present here is not exclusively "Canadian", but we hope that through the new journal MSI, we can contribute to scientific dissemination of knowledge and add Canadian values to the scientific discussion.

The next issue of MSI is planned for the fall of 2020 and is expected to appear again as a special issue of AIMS Mathematics.

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