INTERESTING DETERMINANTS AND INVERSES OF SKEW LOEPLITZ AND FOEPLITZ MATRICES

Qingyan Meng; Xiaoyu Jiang; Zhaolin Jiang

doi:10.11948/20210070

2021 Volume 11 Issue 6

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Qingyan Meng, Xiaoyu Jiang, Zhaolin Jiang. INTERESTING DETERMINANTS AND INVERSES OF SKEW LOEPLITZ AND FOEPLITZ MATRICES[J]. Journal of Applied Analysis & Computation, 2021, 11(6): 2947-2958. doi: 10.11948/20210070

Citation:

Qingyan Meng, Xiaoyu Jiang, Zhaolin Jiang. INTERESTING DETERMINANTS AND INVERSES OF SKEW LOEPLITZ AND FOEPLITZ MATRICES[J]. Journal of Applied Analysis & Computation, 2021, 11(6): 2947-2958. doi: 10.11948/20210070

INTERESTING DETERMINANTS AND INVERSES OF SKEW LOEPLITZ AND FOEPLITZ MATRICES

1.
School of Mathematics and Statistics, Linyi University, Linyi, 276000, China
2.
School of Information Science and Technology, Linyi University, Linyi, 276000, China

Corresponding authors: Email: jxy19890422@sina.com(X. Y. Jiang); Email: jzh1208@sina.com(Z. L. Jiang)
Fund Project: The authors were supported by Natural Science Foundation of Shandong Province (Nos. ZR2020MA051, ZR2020QA035) and the PhD Research Foundation of Linyi University (No.LYDX2018BS052)

Abstract

Abstract

In this paper, we show that there is an intimate relationship between Toeplitz matrix, tridiagonal Toeplitz matrix, the Fibonacci number, the Lucas number, and the Golden Ratio. We introduce skew Loeplitz and skew Foeplitz matrices and derive their determinants and inverses by construction. Specifically, the determiant of $ n\times n $ skew Loeplitz matrix can be expressed by the $ (n+1) $st Fibonacci number. The inverse of skew Loeplitz matrix is sparse and can be expressed by the $ n $th and $ (n+1) $st Fibonacci numbers. Similarly, the determinant of $ n\times n $ skew Foeplitz matrix also can be expressed by the $ (n+1) $st Lucas number. The inverse of skew Foeplitz matrix can be expressed by only seven elements with each element being the explicit expression of the Lucas or Fibonacci numbers. We also calculate the determinants and inverses of skew Lankel and skew Fankel matrices.
- determinant /
- inverse /
- Fibonacci number /
- Lucas number /
- skew Foeplitz matrix /
- skew Loeplitz matrix
MSC: 15A09, 15A15, 65F50

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References

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