本文關(guān)鍵詞：矩陣奇異值及酉不變范數(shù)的研究　出處：《陜西師范大學(xué)》2015年碩士論文　論文類型：學(xué)位論文

  更多相關(guān)文章： 奇異值 奇異值分解 酉矩陣 極分解 弱受控 Hadamard積 半正定2×2塊矩陣 酉不變范數(shù)

【摘要】：矩陣?yán)碚摬坏哂胸S富的研究內(nèi)容,又是一門最有實(shí)用價(jià)值的數(shù)學(xué)工具.在矩陣?yán)碚撝?奇異值分解和奇異值不等式在高水平的統(tǒng)計(jì)計(jì)算和酉不變范數(shù)理論等中起著重要的作用.多年來,許多專家學(xué)者從不同的角度用不同的方法對(duì)有關(guān)奇異值的問題進(jìn)行了很多的研究和推廣,拓展了奇異值的應(yīng)用范圍和理論體系.本文用不同的方法對(duì)奇異值分解和Weyl定理進(jìn)行了證明,并給出了一些推論.同時(shí)酉不變范數(shù)作為矩陣?yán)碚摰囊粋�(gè)重要數(shù)值特征,在矩陣計(jì)算、最佳逼近問題及擾動(dòng)理論中有重要的作用.此外,分塊矩陣作為一種特殊的矩陣,是探討矩陣內(nèi)在性質(zhì)的重要工具,本文將半正定2×2分塊矩陣和酉不變范數(shù)結(jié)合起來,得到了一系列矩陣的酉不變范數(shù)不等式.本文共分三章,具體內(nèi)容如下：第一章主要介紹了本文中涉及的一些符號(hào)概念,主要有正規(guī)矩陣、酉矩陣、Hadamard積、弱受控、log-弱受控、壓縮矩陣、半正定矩陣的半正定平方根、矩陣的廣義逆,同時(shí)又介紹了一些已知的結(jié)論和定理.第二章首先給出了奇異值分解定理,然后分別從譜分解和極分解的角度給出了奇異值分解的其它證明方法,并由此得出了主對(duì)角元向量和奇異值向量的弱受控關(guān)系,并給出了Weyl定理的其它證明及推論.第三章利用酉不變范數(shù)和分塊矩陣的一些性質(zhì),將酉不變范數(shù)和半正定2×2分塊矩陣結(jié)合起來,從而得到了一系列關(guān)于矩陣酉不變范數(shù)的不等式.
[Abstract]:Matrix theory not only has abundant contents, and is a mathematical tool for a most practical value. In matrix theory, singular value decomposition and singular value inequalities in the high level of statistical calculation and unitarily invariant norm theory plays an important role in. Over the years, many experts and scholars from different angles with different the method of the singular value problems were studied and many marketing, expand the scope of application of the singular values and theoretical system. We use a different method of singular value decomposition and Weyl theorem are proved, and some inference. At the same time unitarily invariant norm as an important numerical characteristics of matrix theory. In the matrix, optimal approximation problem and perturbation theory has an important role. In addition, the block matrix is a special matrix, is an important tool to study the intrinsic properties of the matrix, semi positive definite 2 x 2 points Block matrix and unitary invariant norm together, obtained a series of matrix unitarily invariant norm inequalities. This paper is divided into three chapters, the specific contents are as follows: the first chapter mainly introduces some symbolic concepts involved in this paper, the main normal matrix, unitary matrix, Hadamard product, log- controlled by the weak, weak control, matrix compression, positive semidefinite matrix positive semidefinite square root, generalized inverse matrix, and introduces some known results and theorems. The second chapter gives the singular value decomposition theorem, and then from the spectral decomposition and decomposition of the angle of the other that singular value decomposition method, and obtained the main diagonal element vector the singular value vector and weak controlled relation, and gives other proof of the Weyl theorem and corollaries. Some properties of the third chapter is the unitarily invariant norm and block matrix, the unitarily invariant norm and semi definite 2 * 2 block matrix combination A series of inequalities about the unitary norm of the matrix are obtained.

【學(xué)位授予單位】：陜西師范大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2015
【分類號(hào)】：O151.21

【參考文獻(xiàn)】

相關(guān)期刊論文 前5條

1 曾誠;;關(guān)于矩陣Schur補(bǔ)的若干矩陣不等式[J];電子世界;2014年18期

2 鄒黎敏;馮玉明;;關(guān)于酉不變范數(shù)的幾個(gè)不等式(英文)[J];華東師范大學(xué)學(xué)報(bào)(自然科學(xué)版);2011年05期

3 鄒黎敏;;矩陣的幾個(gè)不等式[J];數(shù)學(xué)學(xué)報(bào);2012年04期

4 鄭禪;李寒宇;;半定內(nèi)積下的矩陣奇異值分解[J];山東大學(xué)學(xué)報(bào)(理學(xué)版);2014年12期

5 黃介武;;酉不變范數(shù)幾何-算術(shù)平均值不等式的改進(jìn)[J];西南師范大學(xué)學(xué)報(bào)(自然科學(xué)版);2014年08期

，

本文編號(hào)：1415492