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Interesting Explicit Expressions of Determinants and Inverse Matrices for Foeplitz and Loeplitz Matrices

Author

Listed:
  • Zhaolin Jiang

    (School of Mathematics and Statistics, Linyi University, Linyi 276000, China)

  • Weiping Wang

    (School of Mathematics and Statistics, Linyi University, Linyi 276000, China)

  • Yanpeng Zheng

    (School of Automation and Electrical Engineering, Linyi University, Linyi 276000, China)

  • Baishuai Zuo

    (School of Mathematics and Statistics, Linyi University, Linyi 276000, China)

  • Bei Niu

    (School of Mathematics and Statistics, Linyi University, Linyi 276000, China)

Abstract

Foeplitz and Loeplitz matrices are Toeplitz matrices with entries being Fibonacci and Lucas numbers, respectively. In this paper, explicit expressions of determinants and inverse matrices of Foeplitz and Loeplitz matrices are studied. Specifically, the determinant of the n × n Foeplitz matrix is the ( n + 1 ) th Fibonacci number, while the inverse matrix of the n × n Foeplitz matrix is sparse and can be expressed by the n th and the ( n + 1 ) th Fibonacci number. Similarly, the determinant of the n × n Loeplitz matrix can be expressed by use of the ( n + 1 ) th Lucas number, and the inverse matrix of the n × n ( n > 3 ) Loeplitz matrix can be expressed by only seven elements with each element being the explicit expressions of Lucas numbers. Finally, several numerical examples are illustrated to show the effectiveness of our new theoretical results.

Suggested Citation

  • Zhaolin Jiang & Weiping Wang & Yanpeng Zheng & Baishuai Zuo & Bei Niu, 2019. "Interesting Explicit Expressions of Determinants and Inverse Matrices for Foeplitz and Loeplitz Matrices," Mathematics, MDPI, vol. 7(10), pages 1-19, October.
    • Handle: RePEc:gam:jmathe:v:7:y:2019:i:10:p:939-:d:275118
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    References listed on IDEAS

    as
    1. Jiang, Xiaoyu & Hong, Kicheon, 2015. "Explicit inverse matrices of Tribonacci skew circulant type matrices," Applied Mathematics and Computation, Elsevier, vol. 268(C), pages 93-102.
    2. Chen, Hao & Wang, Xiaoli & Li, Xiaolin, 2019. "A note on efficient preconditioner of implicit Runge–Kutta methods with application to fractional diffusion equations," Applied Mathematics and Computation, Elsevier, vol. 351(C), pages 116-123.
    3. Zhaolin Jiang & Yanpeng Gong & Yun Gao, 2014. "Invertibility and Explicit Inverses of Circulant-Type Matrices with -Fibonacci and -Lucas Numbers," Abstract and Applied Analysis, Hindawi, vol. 2014, pages 1-9, May.
    4. Chen, Hao & Zhang, Tongtong & Lv, Wen, 2018. "Block preconditioning strategies for time–space fractional diffusion equations," Applied Mathematics and Computation, Elsevier, vol. 337(C), pages 41-53.
    5. Xiaoyu Jiang & Kicheon Hong, 2014. "Exact Determinants of Some Special Circulant Matrices Involving Four Kinds of Famous Numbers," Abstract and Applied Analysis, Hindawi, vol. 2014, pages 1-12, June.
    6. Juan Li & Zhaolin Jiang & Fuliang Lu, 2014. "Determinants, Norms, and the Spread of Circulant Matrices with Tribonacci and Generalized Lucas Numbers," Abstract and Applied Analysis, Hindawi, vol. 2014, pages 1-9, May.
    7. Zheng, Yanpeng & Shon, Sugoog, 2015. "Exact determinants and inverses of generalized Lucas skew circulant type matrices," Applied Mathematics and Computation, Elsevier, vol. 270(C), pages 105-113.
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