【摘要】：循環(huán)矩陣類是一類具有特殊結(jié)構(gòu)與性質(zhì)的矩陣,近幾年來(lái)對(duì)循環(huán)矩陣的探究已經(jīng)延伸到了各個(gè)方面,并成為了數(shù)學(xué)領(lǐng)域中極其活躍的研究課題.循環(huán)矩陣有著普通矩陣所沒(méi)有的特殊結(jié)構(gòu)及性質(zhì),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(xué)者們都對(duì)其進(jìn)行了系統(tǒng)的研究.本文先對(duì)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(xiàng)式(廣義Fibonacci多項(xiàng)式,第一,二類切比雪夫多項(xiàng)式)的基本定義,理論,性質(zhì)進(jìn)行了具體闡述,然后將著名多項(xiàng)式應(yīng)用到循環(huán)矩陣中對(duì)其行列式和譜范數(shù)進(jìn)行了研究.主要的研究成果有:1.討論了幾個(gè)特殊循環(huán)矩陣的行列式,其一是包含廣義Fibonacci多項(xiàng)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(yùn)用多項(xiàng)式因式分解的逆變換以及這兩類循環(huán)矩陣特殊的結(jié)構(gòu)性質(zhì)和廣義Fibonacci多項(xiàng)式的通項(xiàng)公式,表示出其行列式的顯式表達(dá)式,另一方面是包含Chebyshev多項(xiàng)式的行首加γ尾γ右循環(huán)矩陣的行列式計(jì)算方式.2.研究了 γ-循環(huán)矩陣的包含廣義Fibonacci多項(xiàng)式的范數(shù),由矩陣范數(shù)的概念,通過(guò)一些代數(shù)方法進(jìn)而給出譜范數(shù)的上下界估計(jì).
[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(xué)位授予單位】：西北大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類號(hào)】：O151.21

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