【摘要】：從上世紀(jì)開(kāi)始科研人員就在陸續(xù)討論根據(jù)一些矩陣的特性要素來(lái)重組矩陣問(wèn)題,稱之為矩陣的逆特征值問(wèn)題。逆特征值問(wèn)題源自于量子力學(xué),分子光特征值學(xué),自動(dòng)控制及核工程等眾多領(lǐng)域的實(shí)際應(yīng)用的需要。逆特征值問(wèn)題不僅為眾多領(lǐng)域的研究提供了一個(gè)較為理想的工具和一個(gè)較為滿意的數(shù)學(xué)方法,同時(shí),逆特征值問(wèn)題由于其本身的復(fù)雜性與擾動(dòng)性所帶來(lái)的一種極為吸引人的數(shù)學(xué)魅力,使其具有更加廣闊的探究前景,吸引了眾多學(xué)者研究相關(guān)課題。目前,研究的矩陣多數(shù)情況局限為實(shí)數(shù)情況,對(duì)非零數(shù)為復(fù)數(shù)情況的矩陣的研究尚少。本文的主要內(nèi)容是對(duì)一類特殊形式的復(fù)對(duì)稱矩陣即擬三對(duì)角矩陣的逆特征值問(wèn)題的研究。本文的創(chuàng)新點(diǎn)在于將數(shù)域擴(kuò)大,由研究相對(duì)成熟的實(shí)數(shù)域延拓至復(fù)數(shù)域,并得到較為理想的結(jié)論和較為穩(wěn)定的算法。下面將本文的主要內(nèi)容歸結(jié)為三方面。首先,研究擬三對(duì)角矩陣的特征值問(wèn)題。本文討論了擬三對(duì)角矩陣及其主子陣的特征值情況及其兩者的關(guān)系,我們得到了擬三對(duì)角矩陣的特征值是互異的實(shí)數(shù),其主子陣的特征值也是互異實(shí)數(shù)且兩組特征值滿足交錯(cuò)性。在這部分內(nèi)容中將數(shù)域和重?cái)?shù)細(xì)化,使得逆特征值問(wèn)題的已知條件更具體明晰,給逆特征值問(wèn)題的研究帶來(lái)方便。其次,對(duì)擬三對(duì)角矩陣的逆特征值問(wèn)題進(jìn)行探索。在這部分我們從爪形矩陣的構(gòu)造入手,通過(guò)給定的特征值構(gòu)造爪形矩陣的邊界元素,并得到了擬三對(duì)角矩陣逆特征值問(wèn)題存在的充分條件,在保證解存在的前提下,根據(jù)爪形矩陣?yán)糜舷嗨评碚摌?gòu)造擬三對(duì)角矩陣。最后,給出了擬三對(duì)角矩陣逆特征值問(wèn)題的算法,并給出三個(gè)典型的算例,同時(shí)借助Matlab對(duì)算法進(jìn)行了大量的數(shù)據(jù)驗(yàn)證,數(shù)據(jù)試驗(yàn)表明在滿足解存在的條件下,任意給定數(shù)據(jù)所得到的矩陣符合我們的理想情況,且此算法的穩(wěn)定性較好。
[Abstract]:Since the last century, researchers have been discussing the problem of reconstructing matrices according to the characteristic elements of matrices. It is called the inverse eigenvalue problem of matrices. Research in this field provides an ideal tool and a satisfactory mathematical method. At the same time, inverse eigenvalue problem has a very attractive mathematical charm because of its complexity and perturbation, which makes it have a broader exploration prospect and attracts many scholars to study related topics. The main content of this paper is to study the inverse eigenvalue problem of a special kind of complex symmetric matrix, i.e. Quasi-Tridiagonal matrix. In this paper, the eigenvalue problem of Quasi-Tridiagonal matrices is studied. In this paper, the eigenvalue of Quasi-Tridiagonal matrices and their principal submatrices and their relations are discussed. We obtain that the eigenvalues of Quasi-Tridiagonal matrices are different from each other. In this part, the number field and multiplicity are refined to make the known conditions of inverse eigenvalue problem more clear, which brings convenience to the study of inverse eigenvalue problem. Secondly, the inverse eigenvalue problem of Quasi-Tridiagonal matrix is explored. In this paper, we start with the construction of claw matrix, construct the boundary elements of claw matrix by given eigenvalue, and obtain the sufficient conditions for the existence of inverse eigenvalue problem of Quasi-Tridiagonal matrix. On the premise of guaranteeing the existence of solution, we construct Quasi-Tridiagonal matrix by using unitary similarity theory according to claw matrix. The algorithm of inverse eigenvalue problem is given, and three typical examples are given. At the same time, a large number of data validation of the algorithm is carried out with the help of MATLAB. The data experiments show that the matrix obtained from any given data conforms to our ideal condition and the stability of the algorithm is good.
【學(xué)位授予單位】：哈爾濱工業(yè)大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2015
【分類號(hào)】：O151.21

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本文編號(hào)：2197859