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

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

doi:10.3390/math7100939

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 nth 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.

Highlights

Toeplitz matrices often arise in statistics, econometrics, psychometrics, structural engineering, multichannel filtering, reflection seismology, etc
The authors [10] proposed the invertibility of generalized Lucas skew circulant matrices and provided the determinant and the inverse matrix
Jiang et al [15] gave the invertibility of circulant type matrices with the sum and product of Fibonacci and Lucas numbers and provided the determinants and the inverses of the these matrices

Summary

Introduction

Toeplitz matrices often arise in statistics, econometrics, psychometrics, structural engineering, multichannel filtering, reflection seismology, etc. (see [1,2] and references therein). The authors [10] proposed the invertibility of generalized Lucas skew circulant matrices and provided the determinant and the inverse matrix. Jiang et al [15] gave the invertibility of circulant type matrices with the sum and product of Fibonacci and Lucas numbers and provided the determinants and the inverses of the these matrices. Hong [16] studied exact form determinants of the RSFPLR circulant matrices and the RSLPFL circulant matrices involving Padovan, Perrin, Tribonacci, and the generalized Lucas number by the inverse factorization of a polynomial. We will show the explicit determinants and inverses of the Foeplitz matrix and Fankel matrix both involving Fibonacci numbers (see Definitions 1 and 2 below), and the Loeplitz matrix and Lankel matrix both involving Lucas numbers (see Definitions 3 and 4). We establish the relationship between the determinant of these matrices and Fibonacci or Lucas numbers

Determinant and Inverse Matrix of a Foeplitz Matrix

Determinant and Inverse Matrix of a Fankel Matrix

Determinant and Inverse Matrix of a Loeplitz Matrix

Determinant and Inverse Matrix of a Lankel Matrix

Example