南航矩阵论等价关系

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The study of Equivalence Relations

Abstract

According to some relative definitions and properties, to proof that if B can be obtained from A by performing elementary row operations on A, ~ is an equivalence relation, and to find the properties that are shared by all the elements in the same equivalence class. To proof that if B is can be obtained from A by performing elementary operations, Matrix S A ∈ is said to be equivalent to matrix S B ∈, and ~A B means that matrix S A ∈ is similar to S B ∈, if let S be the set of m m ⨯ real matrices.

Introduction

The equivalence relations are used in the matrix theory in a very wide field. An equivalence relation on a set S divides S into equivalence classes. Equivalence classes are pair-wise disjoint subsets of S . a ~ b if and only if a and b are in the same equivalence class.This paper will introduce some definitions and properties of equivalence relations and proof some discussions.

Main Results

Answers of Q1

(a) The process of the proof is as following,obviously IA=A,therefore ~ is reflexive;we know B can be obtained from A by performing elementary row operations on A,we assume P is a matrix which denote a series of elementary row operations on A.Then ,we have PA=B,(A~B),and P is inverse,obviously we have A=P -1B,(B~A).So ~ is symmetric.We have another matrix Q which denote a series of elementary row operations on B,and the result is C,so we have QB=C.And we can obtain QB=Q(PA)=QPA=C,so A~C.Therefore,~ is transitive. Hence, ~ is an equivalence relation on S .

(b) The properties that are shared by all the elements in the same equivalence class are as followings: firstly,the rank is the same;secondly,the relation of column is not changed;thirdly,two random matrices are row equivalent;fourthly,all of the matrices

can be raduced as ⎥⎦

⎤⎢⎣⎡00X E n

. (c) ⎥⎦

⎤⎢⎣⎡0000 is a representative element for each equivalence class corresponding to rank 0.

⎥⎦

⎤⎢⎣⎡00y x (x and y can not be zero at the same time)is a representative element for each equivalence class corresponding to rank 1.

⎥⎦

⎤⎢⎣⎡1001 is a representative element for each equivalence class corresponding to rank 2.

Answers of Q2

(a)The process of the proof is as following,,obviously IAI=A,therefore ~ is reflexive;we know B can be obtained from A by performing elementary operations on A,we assume P and Q is a matrix which denote a series of elementary operations on

A.Then ,we have PAQ=B,(A~B),and P is inverse,obviously we have A=P -1BQ -1,(B~A).So ~ is symmetric.We have other matrices M and N which denote a series of elementary operations on B,and the result is C,so we have MBN=C.And we can obtain MBN=M(PAQ)N=MPAQN=C,so A~C.Therefore,~ is transitive. Hence, ~ is an equivalence relation on S .

(b)Obviously,the rank haven ’t changed.Secondly,all of the matrices can be raduced as ⎥⎦

⎤⎢⎣⎡000n E (c) ⎥⎦

⎤⎢⎣⎡0000 is a representative element for each equivalence class corresponding to rank 0.

⎥⎦

⎤⎢⎣⎡0001 is a representative element for each equivalence class corresponding to rank 1.

⎥⎦

⎤⎢⎣⎡1001 is a representative element for each equivalence class corresponding to rank 2.

Answers of Q3

(a)The process of the proof is as following,obviously IAI -1=A,therefore ~ is reflexive;