本文選題：張量奇異值 + 高階奇異值分解�。� 參考：《哈爾濱工業(yè)大學(xué)》2017年碩士論文

【摘要】：張量的奇異值及高階奇異值分解(HOSVD)被廣泛地應(yīng)用于多個(gè)學(xué)科之中。近年來,該理論引起了諸多學(xué)者的普遍關(guān)注,是國內(nèi)外專家和學(xué)者研究的熱門課題。本文主要研究張量奇異值及HOSVD具有的性質(zhì)。首先,將正交變換不改變矩陣的奇異值,合同變換保持實(shí)二次型正定性的結(jié)論推廣到張量上。其次,研究矩形張量的奇異值在Kronecker積下具有的性質(zhì),定義張量的Kronecker和,并研究其H-特征值具有的性質(zhì),隨后探究張量的HOSVD在Kronecker積下具有的性質(zhì)。最后,研究具有特殊結(jié)構(gòu)的張量HOSVD具有的性質(zhì),給出并證明3階2維的Hankel張量通過HOSVD得到的核心張量是對(duì)角張量的充分必要條件,以及若干循環(huán)張量具有保結(jié)構(gòu)的HOSVD.
[Abstract]:Zhang Liang's singular value and higher order singular value decomposition (HOSVD) are widely used in many disciplines.In recent years, the theory has attracted the widespread attention of many scholars and is a hot topic for experts and scholars at home and abroad.In this paper, we study the singular value of Zhang Liang and the properties of HOSVD.Firstly, the conclusion that the orthogonal transformation does not change the singular value of the matrix and the contract transformation keeps the positive definiteness of the real quadratic form is extended to Zhang Liang.Secondly, we study the properties of the singular value of rectangle Zhang Liang under Kronecker product, define the Kronecker sum of Zhang Liang, and study the properties of its H-eigenvalue, and then explore the properties of the HOSVD of under the Kronecker product.Finally, the properties of Zhang Liang HOSVD with special structure are studied, and the sufficient and necessary conditions that the core Zhang Liang obtained by HOSVD obtained by HOSVD is the necessary and sufficient condition of the diagonal Zhang Liang, as well as the existence of some conserved Hos SVDs, are given and proved.
【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】：2017
【分類號(hào)】：O183.2

【參考文獻(xiàn)】

相關(guān)期刊論文 前1條

1 羅自炎;祁力群;;半正定張量[J];中國科學(xué):數(shù)學(xué);2016年05期

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本文編號(hào)：1750132