廣義Fibonacci多項式和Chebyshev多項式構成的循環(huán)矩陣的行列式與譜范數(shù)
發(fā)布時間：2018-11-20 10:13
【摘要】：循環(huán)矩陣類是一類具有特殊結構與性質的矩陣,近幾年來對循環(huán)矩陣的探究已經延伸到了各個方面,并成為了數(shù)學領域中極其活躍的研究課題.循環(huán)矩陣有著普通矩陣所沒有的特殊結構及性質,尤其循環(huán)矩陣,γ-循環(huán)矩陣,行首加γ尾γ右循環(huán)矩陣,行尾加γ首γ左循環(huán)矩陣,行斜首加尾右循環(huán)矩陣和行斜尾加首左循環(huán)矩陣這幾類特殊類型的循環(huán)矩陣,眾多學者們都對其進行了系統(tǒng)的研究.本文先對各類循環(huán)矩陣(例如循環(huán)矩陣,γ一循環(huán)矩陣,行斜首加尾右循環(huán)矩陣,行斜尾加首左循環(huán)矩陣,行首加γ尾γ右循環(huán)矩陣,行尾加γ首γ左循環(huán)矩陣),著名的多項式(廣義Fibonacci多項式,第一,二類切比雪夫多項式)的基本定義,理論,性質進行了具體闡述,然后將著名多項式應用到循環(huán)矩陣中對其行列式和譜范數(shù)進行了研究.主要的研究成果有:1.討論了幾個特殊循環(huán)矩陣的行列式,其一是包含廣義Fibonacci多項式的行斜首加尾右循環(huán)矩陣和行斜尾加首左循環(huán)矩陣,主要運用多項式因式分解的逆變換以及這兩類循環(huán)矩陣特殊的結構性質和廣義Fibonacci多項式的通項公式,表示出其行列式的顯式表達式,另一方面是包含Chebyshev多項式的行首加γ尾γ右循環(huán)矩陣的行列式計算方式.2.研究了 γ-循環(huán)矩陣的包含廣義Fibonacci多項式的范數(shù),由矩陣范數(shù)的概念,通過一些代數(shù)方法進而給出譜范數(shù)的上下界估計.
[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 緯 -cyclic matrix, the first and the last 緯 right cyclic matrix, the row end and the 緯 first 緯 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, 緯 -cyclic matrix, oblique first plus tail right cyclic matrix, oblique tail plus first left cyclic matrix, row first plus 緯 -tail 緯 right cyclic matrix, row end plus 緯 first 緯 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 緯 -tail of a row containing Chebyshev polynomials is given. 2. The norm of 緯 -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.
【學位授予單位】：西北大學
【學位級別】：碩士
【學位授予年份】：2017
【分類號】：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 緯 -cyclic matrix, the first and the last 緯 right cyclic matrix, the row end and the 緯 first 緯 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, 緯 -cyclic matrix, oblique first plus tail right cyclic matrix, oblique tail plus first left cyclic matrix, row first plus 緯 -tail 緯 right cyclic matrix, row end plus 緯 first 緯 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 緯 -tail of a row containing Chebyshev polynomials is given. 2. The norm of 緯 -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.
【學位授予單位】：西北大學
【學位級別】：碩士
【學位授予年份】：2017
【分類號】：O151.21
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1 張耀明;塊循環(huán)矩陣方程組的新算法[J];高等學校計算數(shù)學學報;2001年03期
2 吳世s，
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