矩阵论文

Student’s Name: Cai JingStudent’s ID No.: SX1401111College Name: College of Aerospace EngineeringEquivalence RelationsAbstract: In higher algebra, there are many concepts are equivalence relations, such as similarity relation, contract relationship, matrix equivalent. In modern algebra, equivalence relations are more general and abstract, which plays an important role. By applying the general concept in group theory, we can obtained the concept of factor group and coset. And the natural homomorphism is also obtained from the equivalence relation classification. Therefore, equivalence relations have an important meaning in modern algebra, and have a wide range of applications.This project will discuss the following five equivalence relations.Key word: higher algebra; equivalence relation; similarity; contract1. Introduction1.1 Relative Definitions of Equivalence relationsMapping is one way to establish the contact between the two sets, using this link to compare to the two sets. Through the comparison, we can extrapolate properties of another set. In addition, sometimes we need to make a set to divide into several subsets, which are discussed. And the discussion is beneficial to the study of the original collection. There will be introduce the classification of the set and general principles of categorizing collections - equivalence relations.A given binary relation ~ on a set S is said to be an equivalence relation if and only if it is reflexive, symmetric and transitive. Or equivalently, for all a, b and c in S:(1) a ~ a. (Reflexivity)(2) if a ~ b then b ~ a. (Symmetry)(3) if a ~ b and b ~ c then a ~ c. (Transitivity)The equivalence class of a under ~, denoted [a], is defined as=∈.a b S b a[]{|~}An equivalence relation on a set S divides S into equivalence classes. Equivalence classes are pairwise disjoint subsets of S. a ~ b if and only if a and b are in the same equivalence class.1.2 Salient examplesEquivalence relations permeate mathematics with several salient examples readilyavailable:1. Residue classes [a]N consist of all numbers congruent (equivalent) modulo N.2. A negative number is a set of all equivalent pairs (a, b) of integers with a < b,where two pairs (a, b) and (c, d) belong to the same set (equivalence class) if and only if a+d=b+c.3. A rational number is a set of all equivalent pairs (a, b) of integers, where two pairs (a, b) and (c, d) are equivalent if and only if ad = bc.4. An irrational number is an equivalence class of sequences r1, r2, r3, ... of rational numbers, where two sequences {ri} and {si} are equivalent if and only if they there difference converges {ri - si} to 0 as i→∞.2. Several representative equivalence relations2.1 Row equivalence under elementary row operationsIn this part, there are five equivalence relations which will be profoundly discussed. Let S be the set of m n ⨯ real matrices. Matrix S A ∈ is said to be row equivalent to matrix S B ∈ (denoted by ~A B ) if B can be obtained from A by performing elementary row operations on A.2.1.1Proof of equivalence relationFirst, we prove ~ is an equivalence relation on S.Proof :Reflexivity: for the matrix S A ∈, there is E (unit matrix) which EA=A. So reflexivity is satisfied.Symmetry: supposed that P is the product of elementary matrix. Then PA=B, where S A ∈,S B ∈. Because elementary matrix is reversible and the inverse matrix of elementary matrix also is elementary matrix, A=P-1B is obtained. Namely that matrixA can be obtained fromB by performing elementary row operations on B. So symmetry is satisfied.Transitivity:for matrix A,B,C ∈S , then PA=B, QB=C, where P and Q are elementary matrix . Because the product of two elementary matrix also is an elementary matrix, we haveQB=QPA=C, where QP is an elementary matrix.So transitivity is satisfied.Hence ~ is an equivalence relation on S.Then, we can see that the property that is shared by all the elements in the same equivalence class is that the elements are equivalent to each other.The properties that are shared by all the elementsin the same equivalence classare listed as follows ：·All the elements in the same equivalence class have the same row space. That is to say, the dimensions of the column spaces of all the elements under the same equivalence class are the same.·The dimensions of the column spaces of all the elements under the same equivalence class are the same. That is to say, take A, B as any two matrices under this equivalence class, the column vectors of A are linearly independent if and only if the corresponding column vectors of Bare linearly independent.2.1.3 Representative elementsIf S is the set of 22⨯real matrices, S can be divided into 3 equivalence class, the rank of each row space is 0, 1, 2. Thus, a representative element for each equivalence class:123001011,,000001⎛⎫⎛⎫⎛⎫=== ⎪ ⎪ ⎪⎝⎭⎝⎭⎝⎭C C C 2.2Matrix equivalenceunder elementary operationsLet S be the set of m n ⨯ real matrices. Matrix S A ∈ is said to be equivalent to matrix S B ∈ (denoted by ~A B ) if B is can be obtained from A by performing elementary operations.2.2.1Proof of equivalence relationFirst, we prove ~ is an equivalence relation on S.Proof :Reflexivity: for the matrix S A ∈, there is E (unit matrix) which EAE=A. So reflexivity is satisfied.Symmetry: supposed that P and Q are the product of elementary matrix. Then PAQ=B, where S A ∈,S B ∈. Because elementary matrix is reversible and the inverse matrix of elementary matrix also is elementary matrix, A=P-1BQ-1 is obtained. Namely that matrix A can be obtained from B by performing elementary operations onB. So symmetry is satisfied.Transitivity:for matrix A,B,C ∈S , then PAR=B, QBT=C, where P, R, Q and T are elementary matrix . Because the product of two elementary matrix also is an elementary matrix, we haveQBT=QPART=C, where QP and RT are elementary matrix.So transitivity is satisfied.Hence ~ is an equivalence relation on S.The properties that are shared by all the elements in the sameequivalence classare listed as follows ：·All the elements underthe same equivalence class have the same rank.·All the elements underthe same equivalence class the same invariant divisors or determinant divisors.2.2.3 Representative elementsIf S is the set of real matrices, S can be divided into 3 equivalence class, the rank of each class is0, 1, 2. Thus, a representative element for each equivalence class:123001011,,000001⎛⎫⎛⎫⎛⎫=== ⎪ ⎪ ⎪⎝⎭⎝⎭⎝⎭C C C 2.3Equivalencerelations of matrix similarityLet S be the set of m m ⨯ real matrices. ~A B means that matrix S A ∈ is similar to S B ∈.2.3.1Proof of equivalence relationProof :Reflexivity: for the matrix S A ∈, there is E (unit matrix) which EAE-1=A. So reflexivity is satisfied.Symmetry: supposed that P is nonsingular matrix. Then PAP-1=B, where S A ∈,S B ∈. Therefore A=P-1BP is obtained. So symmetry is satisfied.Transitivity:for matrix A,B,C ∈S , then PAP-1=B, QBQ-1=C, where P and Q are nonsingular matrix. Because the product of two elementary matrix also is an nonsingular matrix, we haveQBQ-1=QPA P-1 Q-1=C.So transitivity is satisfied.Hence ~ is an equivalence relation on S.2.3.2Properties sharedThe properties that are shared by all the elements in the same equivalence class are listed as follows ：·The dimensions of all the matrices under the above equivalence class are the same. ·The determinants of all the matrices under the above equivalence class are the same. ·The traces of all the matrices under the above equivalence class are the same.·All the matrices under the above equivalence class have the same eigenvalues. Thus,they have the same characteristic polynomial, and the same primary factors. 2.3.3 Representative elementsenceclass:.100a b ⎛⎫= ⎪⎝⎭C (a, b can be real numbers) 2.4Equivalencerelations of matrixorthogonalsimilarityLet S be the set of m m ⨯ real symmetric matrices. ~A B means that matrix S A ∈ is orthogonally similar to S B ∈.2.4.1Proof of equivalence relationProof :Reflexivity: for the matrix S A ∈, there is E (unit matrix) which EAET=A. So reflexivity is satisfied.Symmetry: supposed that P is an orthogonal matrix. Then PAPT=B, where S A ∈,S B ∈. Because PT=P-1, A=PTBP is obtained. So symmetry is satisfied.Transitivity:for matrix A,B,C ∈S , then PAPT=B, QBQT=C, where P and Q are orthogonal matrix. Because the product of two elementary matrix also is an orthogonal matrix, we haveQBQT=QPAPTQT=C, where QP is an orthogonal matrix.So transitivity is satisfied.Hence ~ is an equivalence relation on S.2.4.2Properties sharedThe properties that are shared by all the elements in the same equivalence class are listed as follows ：·The dimensions of all the matrices under the above equivalence class are the same. ·The determinants of all the matrices under the above equivalence class are the same. ·The traces of all the matrices under the above equivalence class are the same.·All the matrices under the above equivalence class have the same eigenvalues. Thus, they have the same characteristic polynomial, and the same primary factors.2.4.3 Representative elementsIf S is the set of 22⨯ real symmetric matrices, the matrices cannot changeeigenvalues under orthogonal transformations, thus the eigenvalues a, b remainthesame. Arepresentative element for eachequivalence class:100a b ⎛⎫= ⎪⎝⎭C (a, b can be real numbers) 2.5 Equivalence relations of isomorphismLet S be the set of vector spaces over the field R. 12~V V means that 1V is isomorphic to 2V .Let V and W be two vector spaces over the same field F, and let σ：V →W be a linear mapping (also called a homomorphism). σ is called an isomorphism if it is one-to-one. V is said to be isomorphic to W if there exists an isomorphism from V onto W.2.5.1Proof of equivalence relation·Reflexivity: Obviously, 12~V V , since if σ is an orthogonal one-to-one linear mapping ,then 1(V)V σ=,12~V V .·Symmetry: If 12~V V , then there exists an isomorphism σfrom 1V onto 2V ,let 122,y y V ∈,then there exist 121,x x V ∈ ,such that 1122(x )y ,(x )y σσ== ,then:111212(kx )y ,y y (x )k x σσ=+=+, thus:111121212()x x ()()y y y y σσσ---+=+=+11111(k )x ()y k k y σσ--==1σ-is an isomorphism from 2V onto 12,V V is isomorphic to 1V . Hence ,12,V V . ·Transitivity:If 12~V V , 23~V V , then there exists an isomorphism σfrom 1V onto 2V ,and an Isomorphism from 2V onto 3V .Let 122,y y V ∈,then there exist 121,x x V ∈,such that 1122(x )y ,(x )y σσ==. And according to Symmetry, there exist 123,z z V ∈ , such that 1122(),()y z y z ττ==.Thus:12121212(y )(y )(()())()()()()z z x x x x τττσστστσ+=+=+=+1111(ky )((k ))()()kz x x ττστσ===τσ is an isomorphism from 1V onto 3V ,thus 1V is isomorphic to 3V . Hence,13~V V .2.5.2Properties sharedThe properties that are shared by all the elements in the same equivalence class are listed as follows ：·Vector spaces under the above equivalence class have the same rank.·Vector spaces under the above equivalence class, take 1V ，2V as two examples ,then there exists an isomorphism from 1V onto 2V ,let 12,n ξξξ be a basis for 1V ,then 12(),()()n σξσξσξmust be a basis for 2V .2.5.3 Representative elementsA representative vector space for each equivalence class:For m n R ⨯123100010001000000001000100,,000000000010000000000000000m n m n m n⨯⨯⨯⎛⎫⎛⎫⎛⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪=== ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎝⎭⎝⎭⎝⎭C C C (C )i(1i min(m,n)).i rank =≤≤ Conclusion and acknowledgementsEquivalence relations are mainly focusedin this article. Five representative equivalence relationsare discussedthrough the proof of their equivalence relations,finding theproperties that are shared by all the elements in the same equivalence classand a representative element for eachequivalence class.By discussing representativeexamples, we can furtherunderstand the equivalence relations.Thanks for the help of the authors of the references and the help of Mr Cao, our matrix teacher.References[1] Cao Rongmei. Lecture Notes On Matrix Theory.Nanjing:Nanjing University of Aeronautics and Astronautics,2012.[2] Dai Zhenhua.Lecture Notes On Matrix Theory: Sciencep,2000.。