Structured distance to normality of PDDT Toeplitz matrices

Hongxiao Chu; Ziwu Jiang; Xiaoyu Jiang; Yaru Fu; Zhaolin Jiang; Hongxiao Chu; Ziwu Jiang; Xiaoyu Jiang; Yaru Fu; Zhaolin Jiang

doi:10.3934/math.2025846

AIMS Mathematics

2025, Volume 10, Issue 8: 18929-18956. doi: 10.3934/math.2025846

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Research article Topical Sections

Structured distance to normality of PDDT Toeplitz matrices

1.
School of Mathematics and Statistics, Linyi University, Linyi, 276000, China
2.
School of Information Science and Engineering, Linyi University, Linyi, 276000, China
3.
School of Mathematics and Statistics, Taiyuan Normal University, Jinzhong, 030619, China
4.
School of Intelligent Science and Control Engineering, Shandong Vocational and Technical University of International Studies, Rizhao, 276826, China

Received: 06 May 2025 Revised: 29 July 2025 Accepted: 13 August 2025 Published: 20 August 2025
MSC : 15A18, 15A60

Investigating spectral properties and operator-space distance measurements, this research focused on tridiagonal Toeplitz matrices under perturbed Dirichlet boundary conditions (hereafter referred to as PDDT Toeplitz matrices). Explicit analytical expressions for eigenvalues and their associated eigenvectors were derived. These expressions emphasized their critical role in characterizing stability under perturbation conditions. Building on the structural features of PDDT Toeplitz matrices, we developed closed-form solutions to quantify normality distance and departure from normality. Additionally, these solutions analyzed $ \varepsilon $-pseudospectra and eigenvalue sensitivity at both local and collective scales. By evaluating eigenvalue sensitivity through these parameters, our framework further enabled the evaluation of spectral sensitivity within PDDT Toeplitz environments. Through rigorous analysis, it has been shown that a dramatic increase in eigenvalue sensitivity depended exclusively on the magnitude ratio of lower to upper diagonal entries, demonstrating remarkable independence from diagonal terms or the complex phases of non-diagonal elements. The degree to which a matrix deviated from normality could be effectively characterized by examining the absolute values of its sub-diagonal and super-diagonal elements. To conclude, we explored an inverse eigenvalue problem embedded within a constrained optimization framework, which produced trapezoidal PDDT Toeplitz matrices as final optimal computational solutions.
- Toeplitz matrix,
- eigenvalue,
- condition number,
- normal matrix,
- distance,
- Frobenius norm,
- eigenvalue sensitivity,
- inverse eigenvalue problem
Citation: Hongxiao Chu, Ziwu Jiang, Xiaoyu Jiang, Yaru Fu, Zhaolin Jiang. Structured distance to normality of PDDT Toeplitz matrices[J]. AIMS Mathematics, 2025, 10(8): 18929-18956. doi: 10.3934/math.2025846

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Abstract

Investigating spectral properties and operator-space distance measurements, this research focused on tridiagonal Toeplitz matrices under perturbed Dirichlet boundary conditions (hereafter referred to as PDDT Toeplitz matrices). Explicit analytical expressions for eigenvalues and their associated eigenvectors were derived. These expressions emphasized their critical role in characterizing stability under perturbation conditions. Building on the structural features of PDDT Toeplitz matrices, we developed closed-form solutions to quantify normality distance and departure from normality. Additionally, these solutions analyzed $ \varepsilon $-pseudospectra and eigenvalue sensitivity at both local and collective scales. By evaluating eigenvalue sensitivity through these parameters, our framework further enabled the evaluation of spectral sensitivity within PDDT Toeplitz environments. Through rigorous analysis, it has been shown that a dramatic increase in eigenvalue sensitivity depended exclusively on the magnitude ratio of lower to upper diagonal entries, demonstrating remarkable independence from diagonal terms or the complex phases of non-diagonal elements. The degree to which a matrix deviated from normality could be effectively characterized by examining the absolute values of its sub-diagonal and super-diagonal elements. To conclude, we explored an inverse eigenvalue problem embedded within a constrained optimization framework, which produced trapezoidal PDDT Toeplitz matrices as final optimal computational solutions.

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