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

Zhaolin Jiang, Weiping Wang, Yanpeng Zheng, Baishuai Zuo and Bei Niu
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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

Mathematics, 2019, vol. 7, issue 10, 1-19

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.

Keywords: determinant; inverse; Fibonacci number; Lucas number; Toeplitz matrix; Hankel matrix (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2019
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