广义Fibonacci多项式和Chebyshev多项式构成的循环矩阵的行列式与谱范数
发布时间：2018-11-20 10:13
【摘要】：循环矩阵类是一类具有特殊结构与性质的矩阵,近几年来对循环矩阵的探究已经延伸到了各个方面,并成为了数学领域中极其活跃的研究课题.循环矩阵有着普通矩阵所没有的特殊结构及性质,尤其循环矩阵,γ-循环矩阵,行首加γ尾γ右循环矩阵,行尾加γ首γ左循环矩阵,行斜首加尾右循环矩阵和行斜尾加首左循环矩阵这几类特殊类型的循环矩阵,众多学者们都对其进行了系统的研究.本文先对各类循环矩阵(例如循环矩阵,γ一循环矩阵,行斜首加尾右循环矩阵,行斜尾加首左循环矩阵,行首加γ尾γ右循环矩阵,行尾加γ首γ左循环矩阵),著名的多项式(广义Fibonacci多项式,第一,二类切比雪夫多项式)的基本定义,理论,性质进行了具体阐述,然后将著名多项式应用到循环矩阵中对其行列式和谱范数进行了研究.主要的研究成果有:1.讨论了几个特殊循环矩阵的行列式,其一是包含广义Fibonacci多项式的行斜首加尾右循环矩阵和行斜尾加首左循环矩阵,主要运用多项式因式分解的逆变换以及这两类循环矩阵特殊的结构性质和广义Fibonacci多项式的通项公式,表示出其行列式的显式表达式,另一方面是包含Chebyshev多项式的行首加γ尾γ右循环矩阵的行列式计算方式.2.研究了 γ-循环矩阵的包含广义Fibonacci多项式的范数,由矩阵范数的概念,通过一些代数方法进而给出谱范数的上下界估计.
[Abstract]:Cyclic matrix class is a kind of matrix with special structure and properties. In recent years, the research on cyclic matrix has been extended to various aspects and has become an extremely active research topic in the field of mathematics. The cyclic matrix has the special structure and property which the ordinary matrix does not have, especially the cyclic matrix, the 纬 -cyclic matrix, the first and the last 纬 right cyclic matrix, the row end and the 纬 first 纬 left cyclic matrix. Some special types of cyclic matrices, such as the right cyclic matrix and the first left cyclic matrix, have been systematically studied by many scholars. In this paper, we first study all kinds of circulant matrices (such as cyclic matrix, 纬 -cyclic matrix, oblique first plus tail right cyclic matrix, oblique tail plus first left cyclic matrix, row first plus 纬 -tail 纬 right cyclic matrix, row end plus 纬 first 纬 left cyclic matrix). The basic definition, theory and properties of famous polynomials (generalized Fibonacci polynomials, first, two kinds of Chebyshev polynomials) are described in detail. Then, the determinant and spectral norm of famous polynomials are studied by applying them to cyclic matrices. The main research results are as follows: 1. In this paper, the determinants of some special cyclic matrices are discussed. One is the right cyclic matrix with the diagonal first and the tail of the row and the first left cyclic matrix of the row with the generalized Fibonacci polynomial. The explicit expression of determinant is expressed by the inverse transformation of polynomial factorization, the special structural properties of these two kinds of cyclic matrices and the general formula of generalized Fibonacci polynomials. On the other hand, the method of calculating the determinant of the right cyclic matrix with the first and end 纬 -tail of a row containing Chebyshev polynomials is given. 2. The norm of 纬 -cyclic matrix including generalized Fibonacci polynomials is studied. The upper and lower bounds of spectral norm are estimated by some algebraic methods from the concept of matrix norm.
【学位授予单位】：西北大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O151.21
[Abstract]:Cyclic matrix class is a kind of matrix with special structure and properties. In recent years, the research on cyclic matrix has been extended to various aspects and has become an extremely active research topic in the field of mathematics. The cyclic matrix has the special structure and property which the ordinary matrix does not have, especially the cyclic matrix, the 纬 -cyclic matrix, the first and the last 纬 right cyclic matrix, the row end and the 纬 first 纬 left cyclic matrix. Some special types of cyclic matrices, such as the right cyclic matrix and the first left cyclic matrix, have been systematically studied by many scholars. In this paper, we first study all kinds of circulant matrices (such as cyclic matrix, 纬 -cyclic matrix, oblique first plus tail right cyclic matrix, oblique tail plus first left cyclic matrix, row first plus 纬 -tail 纬 right cyclic matrix, row end plus 纬 first 纬 left cyclic matrix). The basic definition, theory and properties of famous polynomials (generalized Fibonacci polynomials, first, two kinds of Chebyshev polynomials) are described in detail. Then, the determinant and spectral norm of famous polynomials are studied by applying them to cyclic matrices. The main research results are as follows: 1. In this paper, the determinants of some special cyclic matrices are discussed. One is the right cyclic matrix with the diagonal first and the tail of the row and the first left cyclic matrix of the row with the generalized Fibonacci polynomial. The explicit expression of determinant is expressed by the inverse transformation of polynomial factorization, the special structural properties of these two kinds of cyclic matrices and the general formula of generalized Fibonacci polynomials. On the other hand, the method of calculating the determinant of the right cyclic matrix with the first and end 纬 -tail of a row containing Chebyshev polynomials is given. 2. The norm of 纬 -cyclic matrix including generalized Fibonacci polynomials is studied. The upper and lower bounds of spectral norm are estimated by some algebraic methods from the concept of matrix norm.
【学位授予单位】：西北大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O151.21
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