矩阵的对角化及其在高等数学中的应用

第１５卷第２期　２０１２年３月　高等数学研究　ＳＴＵＤＩＥＳ　１Ｎ　Ｃ０ＬＬＥＧＥ　ＭＡＴＨＥＭＡＴ１ＣＳ　Ｖｏ１．１５。ＮＯ．２　Ｍａｒ．，２０１２　

一类矩阵的对角化及其应用　

戴丽　

（南京农业大学工学院，江苏南京２１００３１）　

摘要　利用Ｓｙｌｖｅｓｔｅｒ公式，给出一类矩阵对角化的充要条件，由此可得已有相关定理的一种新的证明，并　

对现有文献中的一些结论加以推广．　

关键词　矩阵对角化；特征值；特征向量；Ｓｙｌｖｅｓｔｅｒ公式　

中图分类号　Ｏ１５１　文献标识码　Ａ　文章编号　１００８—１３９９（２０１２）０２—００１７—０３　

矩阵的对角化问题是线性代数及矩阵理论中最　

基本的问题之一，在许多领域都有着重要的应用，且　

一直受到数学工作者的关注．关于矩阵对角化已有　

许多熟知的刻画方法：　阶矩阵Ａ可对角化等价于Ａ　

有　个线性无关的特征向量，又等价于Ａ的特征子　

空间的维数之和为　，等等＿１　］．特别地，我们有定　

理：任意复方阵Ａ可对角化的充要条件是Ａ的最小　

多项式没有重根．　在本文中，我们首先利用Ｓｙｌｖｅｓｔｅｒ公式，给出　

一类矩阵对角化的充要条件．受此启发，我们得到了　

以上定理的新证明．然后，作为应用，我们推广了　

文１－４］中的所有结论．　

作为预备知识，我们首先回顾一个关于矩阵秩　

的著名结论及其推广形式．　

引理１（Ｓｙｌｖｅｓｔｅｒ公式）Ｅ２－ａ］　设Ａ，Ｂ均为７ｌ阶　

方阵，则　

ｒ（Ａ）＋ｒ（Ｂ）≤ｎ＋ｒ（ＡＬ￣）．　

反复应用Ｓｙｌｖｅｓｔｅｒ公式，进一步可证得　

Ｓｙｌｖｅｓｔｅｒ公式的更一般形式．　

引理２　设Ａ　，Ａ　，…，Ａ　均为　阶方阵，则　

ｒ（Ａ１）＋ｒ（ａ２）＋…＋ｒ（Ａ　）≤　

（ｓ一１）　＋ｒ（Ａ１Ａ２…Ａ　：　．　

命题１　对　阶复方阵Ａ，存在自然数Ｓ≥２使　

Ａ　一Ａ　的充要条件是Ａ可以对角化且Ａ的特征值或者是０　

或者是ｓ一１次单位根．　

证明　设存在自然数Ｓ≥２，使：醇　

Ａ　一Ａ．　于是，Ａ的特征值或者是０或者是Ｓ一１次单位根．将　

INTERNATIONAL CONFERENCE ON HARMONISATION OF TECHNICALREQUIREMENTS FOR REGISTRATION OF PHARMACEUTICALS FOR HUMAN USE

Q1DBracketing and matrixing designs for stabilitytesting of drug substances and drug products

Step 2 9 November 2000

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9This draft guidance, when finalized, will represent the Food and Drug Administration's (FDA's) currentthinking on this topic. It does not create or confer any rights for or on any person and does not operateto bind FDA or the public. An alternative approach may be used if such approach satisfies therequirements of the applicable statutes and regulations.INTRODUCTION101.1Objectives of the Guideline1112The objective of this guideline is to provide harmonised guidance on the application of bracketing13and matrixing for stability studies conducted in accordance with principles outlined in the ICH14Q1A Harmonised Tripartite guideline covering Stability Testing of New Drug Substances and15Products (hereafter referred to as the parent guideline).16171.2Background1819Q1A notes that the use of matrixing and bracketing can be applied, if justified, to the testing of20new drug substances and products, but provides no further guidance on the subject.21221.3Scope of the guideline2324This document is an annex to the parent guideline and addresses recommendations for25bracketing and matrixing study designs. Specific principles are provided in this guideline for26situations in which bracketing or matrixing can be applied without further justification. In other27circumstances, bracketing or matrixing is applicable only if further justification is provided.28Sample designs are provided in this guideline for illustrative purposes, and should not be29considered the only, or the most appropriate, designs in all cases.3031322GUIDELINES3435362.1General3738A full study design is one in which samples for every combination of all design factors are tested39at all time points. A reduced design, is one in which samples for every factor combination are not40all tested at all time points. A reduced design can be a suitable alternative to a full design when41multiple design factors are involved in the drug substance or product being evaluated. Any42reduced design should retain the ability to adequately detect differences in stability resulting43from any of the design factors. Before a reduced design is considered, certain assumptions44should be assessed and justified. The potential risk should be considered of establishing a shorter45shelf life than could be derived from a full design due to the reduced amount of data collected.4647During the course of a reduced design study, if it becomes apparent that the reduced testing is48no longer appropriate because, for example, the product appears less stable than expected, a49modified design, that either reverts to full testing or to a less reduced testing design, can be50followed. Once the design is reverted, full testing or less reduced testing should be carried out51through the proposed retest period or shelf life.52532.2Applicability of Reduced Designs5455Reduced designs can be applied to the stability study of most types of drug products, when56appropriate. For the study of drug substances, matrixing is of limited utility and bracketing is57generally not applicable.5859Bracketing or matrixing can be applied with or without justification depending on the60circumstances as discussed in detail below. The degree of justification in each of these cases will61depend on the available supporting data on the product. Data variability and product stability, as62shown by supporting data, should be considered when a matrixing design is applied.6364Bracketing and matrixing are reduced designs based on different principles. Therefore, the use of65bracketing and matrixing together in one design should be considered and scientifically justified.6667Reduced designs can be used for formal stability studies if the principles outlined below are68followed.69702.3Bracketing7172As defined in the glossary to the parent guideline, bracketing is the design of a stability schedule73such that only samples on the extremes of certain design factors, e.g., strength, package size, are74tested at all time points as in a full design. The design assumes that the stability of any75intermediate levels is represented by the stability of the extremes tested. Where a range of76strengths is to be tested, bracketing is applicable if the strengths are identical or very closely77related in composition (e.g., for a tablet range made with different compression weights of a78similar basic granulation, or a capsule range made by filling different plug fill weights of the79same basic composition into different size capsule shells). Bracketing can be applied to different80container sizes of or different fills in the same container closure system.8182

第２２卷　２００８，止　第４期　１１月　五邑大学学报（自然科学版）　ＪＯＵＲＮＡＬ　ＯＦ　ＷＵＹＩ　ＵＮＩＶＥＲＳＩＴＹ　ｆ　Ｎａｔｕｒａｌ　Ｖｂ１．２２　ＮＯ．４　ＮＯＶ．　２００８　

文章编号：１００６．７３０２（２００８）０４—００１０－０６　

扩展联合对角化算法及其在语音分离中的应用　

江　川。　（五邑大学　信息学院，　江太辉　广东　江ｆ１　５２９０２０）　

摘要：用扩展联合对角化（ＪＡＤＥ）算法对混合语音信号进行盲源分离，并对分离前后的语音　信号的波形和频谱进行了分析比较．实验表明，ＪＡＤＥ算法在混合语音信号的分离中是有效的．　关键词：盲信号处理；盲源分离；扩展联合对角化算法；语音信号分离　中图分类号：ＴＮ９１２．３　文献标识码：Ａ　

Ｔｈｅ　Ｊｏｉｎｔ　Ａｐｐｒｏｘｉｍａｔｉｖｅ　Ｄｉａｇｏｎａｌｉｚａｔｉｏｎ　ｏｆ　Ｅｉｇｅｎｍａｔｒｉｘ　Ａｌｇｏｒｉｔｈｍ　ａｎｄ　ｉｔｓ　Ａｐｐｌｉｃａｔｉｏｎ　ｉｎ　Ｓｐｅｅｃｈ　Ｓｅｐａｒａｔｉｏｎ　

ＪＩＡＮＧ　Ｃｈｕａｎ，ＪＩＡＮＧ　Ｔａｉ—ｈｕｉ　（Ｓｃｈｏｏｌ　ｏｆ　Ｉｎｆｏｒｍａｔｉｏｎ，Ｗｕｙｉ　Ｕｎｉｖｅｒｓｉｔｙ，Ｊｉａｎｇｍｅｎ　５２９０２０，Ｃｈｉｎａ）　

Ａｂｓｔｒａｃｔ：Ｔｈｅ　ｊｏｉｎｔ　ａｐｐｒｏｘｉｍａｔｉｖｅ　ｄｉａｇ０ｎａｌｉｚａｔｉ０ｎ　ｏｆ　ｅｉｇｅｎｍａｔｒｉｘ　ｉｓ　ｉｎｔｒｏｄｕｃｅｄ　ｉｎ　ｔｈｉｓ　ｐａｐｅｒ　ａｎｄ　ｓｏｍｅ　ｅｘｐｅｒｉｍｅｎｔｓ　ｏｎ　ｂｌｉｎｄ　ｓｏｕｒｃｅ　ｓｅｐａｒａｔｉｏｎ　ｏｆ　ｍｉｘｅｄ　ｖｏｉｃｅ　ｓｉｇｎａｌｓ　ａｒｅ　ｍａｄｅ．Ｔｈｅ　ｗａｖｅｆｏｒｍ　ａｎｄ　ｓｐｅｃｔｒｕｍ　ｏｆ　ｔｈｅ　ｓｐｅｅｃｈ　ｓｉｇｎａｌｓ　ｂｅｆｏｒｅ　ａｎｄ　ａｆｔｅｒ　ｓｅｐａｒａｔｉｏｎ　ａｒｅ　ｃｏｍｐａｒｅｄ　ａｎｄ　ａｎａｌｙｚｅｄ．Ｅｘｐｅｒｉｍｅｎｔｓ　ｓｈｏｗ　ｔｈａｔ　ＪＡＤＥ　ｉｓ　ａｎ　ｅｆｆｅｃｔｉｖｅ　ａｌｇｏｒｉｔｈｍ　ｆｏｒ　ｔｈｅ　ｂｌｉｎｄ　ｓｏｕｒｃｅ　ｓｅｐａｒａｔｉｏｎ　ｏｆ　ｔｈｅ　ｍｉｘｅｄ　ｓｐｅｅｃｈ　ｓｉｇｎａｌｓ．　Ｋｅｙ　ｗｏｒｄｓ：ｂｌｉｎｄ　ｓｉｇｎａｌ　ｐｒｏｃｅｓｓｉｎｇ；ｂｌｉｎｄ　ｓｏｕｒｃｅ　ｓｅｐａｒａｔｉｏｎ（ＢＳＳ）；ＪＡＤＥ　ａｌｇｏｒｉｔｈｍ；ｓｐｅｅｃｈ　ｓｉｇｎａｌ　ｓｅｐａｒａｔｉｏｎ　

矩阵对角化matlab

矩阵对角化是一种重要的数学运算，它在许多领域中都有着广泛的应用。在matlab中，也可以使用一些函数来进行矩阵对角化操作。本文将介绍矩阵对角化的概念、方法以及在matlab中如何进行矩阵对角化。

一、矩阵对角化的概念

1.1 矩阵对角化的定义

矩阵对角化是指将一个矩阵转换为一个相似的对角矩阵的过程。相似矩阵是指具有相同特征值和特征向量的两个矩阵。

1.2 矩阵相似

两个n×n矩阵A和B称为相似矩阵，如果存在一个可逆矩阵P，使得B=P−1AP。

二、如何进行矩阵对角化

2.1 判断是否可对角化

判断一个n×n方阵A是否可对角化，需要满足以下条件：

（1）A有n个线性无关的特征向量；

（2）A与其它任何一个方阵B相似时，B也可被对角化。

2.2 求解特征值和特征向量

若方针A可对角化，则存在一个可逆矩阵P，使得P−1AP=D，其中D是对角矩阵。因此，我们只需要求解A的特征值和特征向量，就可以得到P和D。

在matlab中，可以使用eig函数来求解方针A的特征值和特征向量。例如：

A=[1 2 3;4 5 6;7 8 9];

[V,D]=eig(A)

其中V表示特征向量矩阵，D表示特征值矩阵。

2.3 对角化

根据上述公式，我们可以得到：

A=PD P−1

因此，我们只需要求出可逆矩阵P和对角矩阵D即可完成对角化操作。

在matlab中，可以使用diag函数来构造对角矩阵。例如：

D=diag([1,2,3]);

同时，我们也可以通过计算特征向量来构造可逆矩阵P。例如：

P=[V(:,1) V(:,2) V(:,3)];

最终，我们就可以得到对角化后的结果：

A=PDP−1

三、matlab中的实例演示

下面以一个具体的例子来演示如何在matlab中进行矩阵对角化操作。

假设有一个方针A：

A=[4 -1 -1;-1 4 -1;-1 -1 4];

首先，我们需要求解该方针的特征值和特征向量。在matlab中，可以使用eig函数来求解：