INTERESTING DETERMINANTS AND INVERSES OF SKEW LOEPLITZ AND FOEPLITZ MATRICES

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. </abstract>