師白娟
(西北大學(xué)數(shù)學(xué)學(xué)院,陜西 西安 710127)
包含切比雪夫多項式的循環(huán)矩陣行列式的計算
師白娟
(西北大學(xué)數(shù)學(xué)學(xué)院,陜西 西安 710127)
行首加r尾r右循環(huán)矩陣和行尾加 r首r左循環(huán)矩陣是兩種特殊類型的矩陣,這篇論文中就是利用多項式因式分解的逆變換這一重要的技巧以及這類循環(huán)矩陣漂亮的結(jié)構(gòu)和切比雪夫多項式的特殊的結(jié)構(gòu),分別討論了第一類、第二類切比雪夫多項式的關(guān)于行首加r尾r右循環(huán)矩陣和行尾加r首r左循環(huán)矩陣的行列式,從而給出了行首加r尾r右循環(huán)矩陣和行尾加r首r左循環(huán)矩陣的行列式顯式表達式.這些顯式表達式與切比雪夫多項式以及參數(shù)r有關(guān).這一問題的應(yīng)用背景主要在循環(huán)編碼,圖像處理等信息理論方面.
行首加r尾r右循環(huán)矩陣;行尾加r首r左循環(huán)矩陣;第一類切比雪夫多項式;第二類切比雪夫多項式;行列式
循環(huán)矩陣類在許多學(xué)科中有很重要的應(yīng)用[1-11],例如圖像處理,通信,信號處理,編碼,預(yù)處理等.P.Davis和江兆林教授已經(jīng)為其研究奠定了深厚的基礎(chǔ).近幾年內(nèi)循環(huán)矩陣的探究已經(jīng)延伸到很多方面,成為了活躍的研究課題.循環(huán)矩陣是其另外的自然延伸,有廣泛的應(yīng)用,特別是在廣義循環(huán)碼方面.xn-rx-r-循環(huán)矩陣被稱為行首加r尾r右循環(huán)矩陣,簡記為RFPrLrR循環(huán)矩陣,比一般的f(x)-循環(huán)矩陣有更好的結(jié)構(gòu)和性質(zhì),所以求解RFPrLrR循環(huán)線性系統(tǒng)有更好的快速算法.
在這篇論文中,主要考慮切比雪夫多項式的關(guān)于行首加r尾r右循環(huán)矩陣和行尾加r首r左循環(huán)矩陣的行列式.由切比雪夫多項式的特征給出了行列式的顯式表達式,這里所運用的技巧正是多項式因式分解的逆變換.首先,我們介紹了行首加r尾r右循環(huán)矩陣和行尾加r首r左循環(huán)矩陣的定義和切比雪夫多項式的特征性質(zhì);然后,我們呈現(xiàn)出主要的結(jié)果和詳細(xì)過程.
先考慮第一類切比雪夫多項式Tn的關(guān)于行首加r尾r右循環(huán)矩陣,行尾加r首r左循環(huán)矩陣的行列式,主要結(jié)論如下:
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Determinants of RFPrLrR circulant matrices of the Chebyshev polynomials
Shi Baijuan
(College of Mathematics,Northwest University,Xi′an 710127,China)
In this paper,two new kind of circulant matrices,i.e.,the RFPrLrR circulant matrix and the RLPrFrL circulant matrix over the complex field C are considered respectively.The determinants of RFPrLrR circulant matrices and RLPrFrL circulant matrices of the Chebyshev polynomials are given by using the inverse factorization of polynomial.The calculation problem of a class determinant involving Chebyshev Polynomials are solved by using the combinatorial method and algebraic manipulations.
Chebyshev polynomials,RFPrLrR circulant matrix,RLPrFrL circulant matrix,determinant
O177.91
A
1008-5513(2016)03-0305-13
10.3969/j.issn.1008-5513.2016.03.009
2016-02-26.
國家自然科學(xué)基金(11371291).
師白娟(1992-),碩士生,研究方向:數(shù)論.
2010 MSC:60B12