A Study On The Power And The End - Toothed Toeplitz Matrices Of Some Circulant Matrices

In this paper, we mainly study the following contents:the determinants and inverse matrices of Gaussian Fibonacci circulant type, skew-circulant type and r circulant matrices, the arbitrary positive integer powers for general r circulant matrices, the inverses of Row First-Plus-Last (RFPL) Toeplitz matrices and solution of RFPL Toeplitz systems of linear equations. The main contents of this paper are arranged as follows.In chapter one, we first introduce the application background of circulant matrices, r circulant matrices and its powers, and Toeplitz matrices, and research status of them at home and abroad. Then we present some preliminaries, such as the definitions of Gaussian Fibonacci r circulant and r left circulant matrices, the definition and upper bounds of spread, the definition of RFPL Toeplitz matrices, which are prepared for the later research. Finally, we state the main work of this paper.In chapter two, by constructing the transformation matrices, we give the expressions for determinants and inverse matrices of Gaussian Fibonacci circulant type matrices, including circulant matrices, left circulant matrices, and g-circulant matrices, and prove the results. Combining with some properties of Fibonacci sequence and Gaussian Fibonacci sequence, we consider the upper bounds of spreads of Gaussian Fibonacci circulant and left circulant matrices. Then using the similar approach, we present the determinants of Gaussian Fi-bonacci skew-circulant and r circulant matrices, and compute their inverses after discussing their invertibility. At last, we provide a method to compute the arbitrary positive integer powers for general r circulant matrices via the combination of the Multinomial Theorem and basic r circulant matrix.In chapter three, we mainly discuss the Row First-Plus-Last (RFPL) Toeplitz matrices. Through constructing the displacement of the structure, we compute the inverses of RFPL Toeplitz matrices. Furthermore, we obtain the inverses of RFPL Hankel matrices by the relationship between RFPL Toeplitz matrices and RFPL Hankel matrices. Next, based on the construction of a different kind of displacement of the structure, the properties for the rank of displacement and LU factorization of matrix, an algorithm for the solution of RFPL Toeplitz systems of linear equations is given.In chapter four, we summarize the main ideas and contents in this paper and give some constructive opinions.