Journal Paper Format

International Journal of Multimedia and Ubiquitous Engineering Vol. 9, No. 11 (2014), pp. 61-74 http://dx.doi.org/10.14257/ijmue.2014.9.11.07

A New Explanation of Relation between Matrices

Li Fengxia Department of Information Science, Heilongjiang International University, Harbin 150025, China [email protected]

Abstract It is known that if two matrices are of same size, there may be the equivalent, similar or congruent relations between matrices. This paper has mainly launches the research about relations between matrices of the same size, different size by matrix operations. Such as there is an association between matrix and its adjoint matrix, adjoint matrix is linear combination of power of A. Using a mathematical formula to unify the three equivalence relations. Keywords: matrix relations classification matrix operations

1. Introduction Only.ILLEGAL. About the relationship between matrices , most of the literature focuses on the equivalence relation , similar relation, congruent relation , for example , is[1]-[6] has discussed the three important relations in the matrix : equivalence , similarity , congruence , and analyzed the difference and connection between them . [7] has induced and analyzed the differences between matrix , the number of operations , the internalfile relations of matrix operations and applications of matrix operations in the determinant of a matrix , rank of matrix , the relation between matrices , elementary transformationsVersion . Because three relations are equivalence relations and matrices can be classified basing on any kind of equivalence relation. Three relations mentioned above can be defined bythis a matrix multiplication operation as follow. Let A , B be matrices of the same type , A=PBQ , where P , Q are invertible matrices , then A is equivalent to B, Further , if A ,by B are square matrices of the same order , Q is the inverse matrix of P , then A is similar to B . If A, B are real symmetric matrices of the same order, Q is the transpose Onlineof P, then A is congruent to B. Since three relations above depends on the multiplication operation of matrix, then whether other relations between matrices can also be given by matrix operations, whether there is an association between matrix and its transpose matrix, adjoint matrix .what relation between two matrices of differentmade row (column ) This paper has mainly launches research about relations between matrices of the same size, different size by matrix operations. Such as the relation between two matrices of same column.

10 11  Book AB, 1 2 .  22  11 2. Preparation of Knowledge Definition 1: A matrix A is said to be equivalent to a matrix B if there exist invertible matrix P, Q, such that PAQ=B.

Definition 2: A matrix A is said to be similar to a matrix B if there exists an invertible matrix P such that PAPB 1  . Definition 3: A matrix A is said to be congruent to a matrix B if there exists an invertible matrix P such that PAPBT  . Definition 4: A relation is said to be an equivalence relation if the relation satisfy reflexivity, symmetricity and transitivity. It is easy to see that the equivalence, the similarity and the congruence of matrices are three equivalence relations. Reflexivity, symmetricity and transitivity, by these three properties, all matrices are classified into classe. Every class has a representative elemen, all matrices in the same class are equivalent to each other, and matrices in different class are not equivalent.

3. Main Results

3.1. Relation between Matrix with its Transpose and Adjoint Matrix.

When mn , let AF mn , supposed mn , k is nonzero number, then

T kA   kA T , RARAT     , R kA  R  A  . (1)

When mn , Let AF nn , A T is the transpose matrixOnly. of A , andILLEGAL. A * is the adjoint matrix of A, E is an identity matrix of order n , then there is a very important equation about A and n is ** , such that AAAAAEn . Some property of the transpose matrix or adjoint matrix A * of A may depend on the original matrix A. Such as, let k a nonzero number. file VersionATn A, kA k A (2) this  n, R A   n  * n 1 * A A n 2  , R A  1, R A   n  1 . (3) by   0 , R A   n 1 Theorem 1: OnlineLet A be a square matrix of order n, then adjoint matrix of A is linear combination of power of A. In particular, when A is invertible, then A made* nn1 2 A   A  ann11 A      a E  (4) a 0 nn1 Where f    Enn  A   a 1      a1  a 0 is the eigenpolynomial of A. Proof: Let A be a square matrix of order n, the following discussion according to the rank of matrixBook A from three aspects. When R A   n , A is a matrix of full rank, and A is an invertible matrix.

nn1 f    E  A    a n 1      a1  a 0 (5) It is the eigenpolynomial of A. By Hamilton-Cayley theorem fA   0 , that is to say

nn12 A A an 1 A      a1 E  a0 E  0 (6)

Because A is invertible, then f00    A  a 0  , and

 1 nn12a n 1 a 1 AA   A     EE  (7) a a a 0 0 0 Then

 1 1 nn1  2 A   A  an 11 A      a E  (8) a 0 1 and AA 1* , A That is A n 1

n  2 A  (9) * AAnn1 2  A   A  ann1 A     a1 E    1,a1 ,   , a 1 , a 0   aa 00A E   0 So A * is the polynomial of A. Only.ILLEGAL. When R A   n 1 , let ,,,   be a basis of linear space V, AA, * are the matrix of 12 n is linear transformation with respect to the basis12,,,   n . Because f A   Ann  a A1      a A is an annihilator polynomial of A, then there n 11file exist B1 , polynomial of A, such that BAB110, 0 . One side, because Version, so let AAA,,,   be independent vectors. 1 2 n 1 AVLAAA ,,,     .But thisare independen, then there is any vector   1 2 n 1  i which can not be linear represented by . Suppose it is  , let B be linear by 1 1 transformation of corresponding to matrix B1 , then B11 is not zero , but AB11  0 , B11 is in A  1  0  . Online * * On the other hand, AA  1  0 , then A  1 is also in , so there is number k, such

* that A kB . Because The role of linear transformation kB , A * in 1 1 1 made 1 * are all zero . And are independent, A k B1 . So is the polynomial of A. When R A   n , RA*   0 , that is A *  0 , So is a zero polynomial and the polynomial of A . Book 11 Example 1: For A  .  23 A 50 , then A is invertible. One hand,

AA 31 * 1 1 2 1  A . (10) AA 1 2 2 2 21 On the other hand, the eigenpolynomial of A of order 2 is

 11 f    E  A  2 45   (11) 23 

Where aa10 4,  5 . Then

A 5 31 A *  A2 1  a E  AE 4   1 2  2  . (12) 21 a 0 5  Moreover some property of the transpose matrix or adjoint matrix of A may inherit from the original matrix A, for example symmetry, invertible, the positive definiteness, orthogonality.

3.2. Relation between two Matrices by their Size Row number, column number are two important indexOnly. of matricesILLEGAL. , Many of the properties of matrix relate to their indexes , such as the rank of a matrix , standard form of matrix etc . is 3.2.1. When two Matrices have the Same Size of m Row and n Column. Suppose mn ABF,  . mn , suppose that mn , A , B havefile the same size , we can only consider the equivalent matrices . Let A, B be matrices of the same size, we can think about whether two matrices are equivalent. Version Proposition 1: Two mn matrices are equivalent if and only if they have the same rank. Based on the equivalent relation, matricesthis can be divided into class m+1= min {m, n}+1 ,

 EE12   O ,  by,   ,...,,EO  (13)  OO   m n m Online m1    n  1    m2    n  2   as the representative element . m = n, A and B are square matrices , We can consider the similar or congruent relations . When two matricesmade are similar. Proposition 2: A is similar to B if and only if their eigenmatrices  EAn  and  EBn  are equivalent. Proposition 3: Let A and B be nn matrices over a number field. The following conditions are equivalent. A Bookand B are similar. A and B have the same determinant divisors. A and B have the same invariant factors. A and B have the same elementary divisors. Proposition 4: Every matrix A is similar to a block diagonal matrix over C which consists of Jordan blocks down the diagonal. Moreover, these blocks are uniquely determined by A up to order.

Proposition 5: Let ABC,  mn , A is similar to B if and only if they have the same Jordan canonical form. A is similar to B if and only if heir elementary divisors collection nn 1 k  ,ni , 1, ...,   F  1 ,...,    k  are the same completely, where ii k, but i has infinite, so based on the similar relation, matrices in number field can be divided into infinite class . When two matrices are congruent. Proposition 6: Two real symmetric matrices are congruent over R if and only if two of the followings are satisfied: They have the same rank; They have the same positive index; They have the same negative index; They have the same signature. Based on the congruent relation, symmetric matrices in Real number field can be divided (nn 1)( 2 ) into1+2+3+…+ (n+1)= class . 2 1  11     O ,  ,  Only., 1 ILLEGAL.  OO   n1    n  1  n1    n  1      O is nn22     

      11 E p       1 , 1 file,..., E      rp  (14)  O   O   O   n2    n  2    Versionn2    n  2    n r    n  r   as the representative element . this Three relations mentioned above can be defined by a matrix multiplication operation. Theorem 2: Let A, B be matrices of the same type, A=PBQ , where P , Q are invertible matrices , then A is equivalent to Bby , further , if A, B are square matrices of the same order , Q is the inverse matrix of P , then A is similar to B . If A, B are real symmetric matrices of the same order, OnlineQ is the transpose of P , then A is congruent to B . 1 1   1 0  Example 2: For AB ,   . made2 2   0 4  Because  1 0   1 0   1 1   1 1   1 0            (15) 0 1   2 1   2 2   0 1   0 4  Then A is equivalent to B, and the product decomposition of the equation is not unique. TheirBook eigenmatrices are not equivalent if and only if A is not similar to B   11   10  EA,  EB (16) 22 04  B is a real symmetric matrix and A isn’t symmetric, so A is not congruent to B. The matrix classification

Matrices Matrices of same size

Square matrices

Figure 1. Venn Diagram of the Set of Matrices

The matrix will be divided into two classes in Figure 1: the same size of matrix and different size matrix . Teaching material mainly research the same size of matrices.

Matrices of same size

congruence equivalence Only.ILLEGAL. similarity is

Figure 2. Venn Diagram of the Set offile Matrices of Same Size In the same size of matrices inVersion Figure 2 can be divided into the matrix of equivalence , similarity , congruence , these three kinds of relations are equivalence relations, satisfying reflexivity , symmetricity and transitivity . this In particular, if the two square matrices of same order are similar, then they must be equivalent. In the same way, if the two square matrices of same order are congruent, then they are equivalent, but if two matrix by are equivalent, but they are not necessarily similar or congruent. Online 3.2.2. When there is one index and only one of the same. Let A Fs n ,, B  Ft n s  t , A A  tn  suppose st , it existsmade a matrix C , such that  F , and B have the same size. C C A Proposition 7: If R R() B n , and B are the full column rank , then C E Book E n n  can be written as P =QB=  , where P , Q are invertible matrices . it O O t n  n t n  n contains that = P  1 QB .

10 11  Example 3: Let AB, 1 2 .  22  11 A A Because RRB( ) 2 , then and B are the full column rank , suppose C C 11 A  C a, b such that  22,    C  ab 1 0 0  1 1 0   1 0 0  1 1   1 0    1       0 1 00 021022 01 (17)           4        0 (a  b ) 1    a 0 1  a b  0 0  0 0 1 10010010010         10            001010 11012 01 (18)      Only.            ILLEGAL.   0100    21  10111 is   00  E 2 Then may be written as P =QB= , which O 12 file 11 0 1 0 0 24  1 1 0  Version 1 0 0     1   1 1 P  0 1 0 0 0 2 1 0   0 (19)   this     4   2 4 0 (a  b ) 1  a 0 1    0 0 1   1 1   (b a )  ( a  b ) 1 by 24 100100100       100  Online        Q  001010 110 1  21 (20)                 010021      101   110  It contains that made  1 11 0 24 11  1 0 0  1 0  11     22= 0 1 2 1 1 2 (21) Book     24     ab  1 1 0 1 1 11      (b a )  ( a  b ) 1 24

A A If R R() B t , and B are the full row rank , then EO, may be written    t t n t   C C as P =BQ= .

B  T TT TTT If R() B R t , then R(), B R B A  t , then matrix equation BXA A has the unique solution, therefore matrix equation XBA has the unique solution.

Proposition 8: Let G be a full column matrix of mn , H be a full row matrix of pq , then for any np matrix A satisfies RGARARAH()()() . Let A be a matrix of rank r, then there are P full column matrix of mr , Q be a full row matrix of rn such that A=PQ. Let A Fm p ,, B  Fm q p  q , it is no longer to be discussed here.

3.2.3. When Row Index, Column Index of two MatricesOnly. are CrossILLEGAL. Correlation Equal Let AFBFs n ,,n t When st , suppose st , then isand B T have the same size. Here it is no longer to be discussed. When st , suppose sn , then A and have the same size, the products AB and BA always make sense and these products are squarefile of sizes s s, n n respectively. But generally equality AB=BA does not hold. Version Proposition 9: this AB and BA have the same trace: tr (AB) = tr (BA). EAB and EBA have the same invertibility : E -AB is invertible if and only if E - s n by s n BA is invertible .

Online ns   EABEBAsn   . (22)

11 made 1 2 2 Example 4: For A  21 , B  .  114 11 2 1 2 71 AB 3 3 0 , BA . (23) Book   54 0 3 6 tr (AB) =11= tr (BA). (24)

1  1  2 61 EAB  3  2 0 , EBA  . (25) 32 53 0 3 5

EAB3  , EBA2  are invertible.

32  EABEBA32     . (26) 3.2.4. When Row Number, Column Number of two Matrices are Completely Different. Let AFBFm n ,,s t where any two of m, n, s, t is unequal , suppose ms , nt , A and B have the same size. C A In particular, when R  RB() , matrix A can be seen as a sub block of matrix B. C A  n>t, when mn st, and BB, 1  have the same size. Here it is no longer to be A 1 discussed. Only.ILLEGAL.

3.3. Relation between Matrix using Matrix Decomposition is Matrix decomposition is an effective tool to realize the large-scale data processing and analysis. As we all know, proposition conditions more special, the conclusion is more perfect, but this conclusion is not commonly used, the followingfile statement mainly focuses on general matrix decomposition structure, in order to get the general, widely used conclusions. Version 3.3.1. Addition Decomposition. If matrixthis A can be decomposed into two or several matrix such as A is sum of some matrix and unit matrix , then we can use the matrix decomposition formula to solve a series of related problems such as matrix and A can be commutative; power of A ; or eigenvalue of A. by Each square matrix of order n is usually uniquely presented by the sum of the symmetric matrix and skew-symmetricOnline matrix. That is for every matrix A=S+T, where S is a symmetric matrix and T is a skew-symmetric matrix. The decomposition gives theory basis and method by using the general matrix to construct symmetric matrices,made skew-symmetric matrices. AAAATT AA T AA T A  , where is a symmetric matrix and is a skew- 22 2 2 symmetric matrix. Each square matrix of order n is usually represented by the sum of the scalar matrix and matrixBook of trace 0. Each square matrix of order n in complex field is usually represented by the sum of the nilpotent matrix B and an diagonal matrix C and BC=CB. Each matrix of rank r is usually represented by the sum of the r-th matrices of rank 1. Each square matrix of order n is usually represented by the linear sum of

T ()j E e eT , w h ere e  0,...,0,1,0,...,0 ,i , j 1,2,..., n ij i j j . (27)  A a F nn A a E Such as ij  , then  ij ij . 1,i j n This decomposition is based on the linear space theory and shows the importance of the matrix of shaped like E ij .

3.3.2. Product Decomposition. If a matrix A can be decomposed into the product of two matrices (such as matrix A of rank 1) , or similarity diagonalization (such as A has n linearly independent eigenvectors) , then the power of A, the polynomial of A and the rank of A can be calculated by using the product decomposition of A. Case 1: Some general product decomposition, such as: Any matrix of rank 1 can be factorized into the product of two vectors of rank 1.  1 2 3

 2014 Example 5: Let A 1  2  3 to calculate A .   2 4 6  1 Only.11ILLEGAL.    T      Because R(A)=1 , A  1  1 2 3 , where1 , 2        is    2 23    AA2   T    T    fileTT     9 (28) AAAA3 2 9 2 , (29) Version 92 0 1 3 2 92 0 1 3 3 9 2 0 1 3

2 0 1 4 2 0 1 3 2 0 1 3 2 0 1 3 2 0 1 3 AA9 9this 2  9  3  9 . (30) 2 92 0 1 3  4  92 0 1 3  6  9 2 0 1 3  Each square matrix A is usually representedby by the product of matrices such as T ()j EOnline a E , where E  e eT , e  0,...,0,1,0,...,0 ,,i j 1,2,..., n . (31) n ij ij ij i j j   Case 2: Some famous special product decomposition, such as triangle decomposition, schur decomposition, fitting decomposition, singular value decomposition, non-negative matrix factorizationmade and so on.

Triangle Decomposition LU Decomposition: Each square matrix A is usually represented by the product of lower triangularBook L and an upper triangular U, such that A=LU. In the calculation of determinant, we often reduced Complex determinant to Upper (lower) triangular determinant using the determinant properties or by row (column) expansion theorem. LU decomposition, each matrix is closely related to upper (lower) triangular matrix.

QR Decomposition: For AC nn , then there exist orthonormal matrix Q, upper triangular matrix R, such that AQR .

1 2 1  Example 6: Let A 1 2 2 , to calculate QR decomposition of A.   0 1 3 1   2   1        Let a  1 ,a  2 , a  2 , use the unit orthogonal method of Schmidt , then 1  2  3         0   1   3  1 1 2 b   1 11 b1 a 1  1,q 1     a 1 (32)  b 22 0 1   0

2 2 3 ba, b 12   2 21

b2 a 2  b1 2, q 2  33  a 2 (33) b, b   b  1 1 1 2 1  3

2 1   2 6 b,, a b a b  1 3  2 3  1 3 Only.2 2 2 2 b a  b  b  , q    ILLEGAL.a  a  a (34) 3 3 1 2 2 3 6 6 1 3 2 3 3 b1,, b 1   b2 b 2  b3  22 2  3 is Make

22  112     660  223 file   22 Q  q,, q q   1 2 ,C   0 1   (35) 1 2 3 Version2 3 6 3 3  1 2 2   2   0 3   00   this33     1 112  220 6 2 0  RC 1  0by1 2  0 3 3 33  (36)  0 0 2 0 0 3 Online 3 2 Therefore QR decomposition of A is

2 112   223 6 20 made  2 AQR  12  0 3 3 . (37) 2 36

1 22 3 0 3 00 3 2 Schur Decomposition: letBookAC nn , then there exist unitary matrix U , such that AUTU H , where T is an upper triangular matrix which diagonal elements are all eigenvalue of A . Fitting Decomposition: Let AF nn , then there exists an invertble matrix T, such that A T d ia g{,} D N T  1 , where D is an invertible matrix, N is a nilpotent matrix.

Square matrix A of order n itself may not be invertble or nilpotent, but A is similar to a Block diagonal matrix consisting of invertble blocks and nilpotent blocks.

Singular Calue Decomposition:

Let AC mn , then A is usually represented by the product of unitary matrix U of order m, semi-positive definite m×n diagonal matrix S and unitary matrix V of order n , such that AUSV * . 1 0 1 Example 7: Let A   ，to calculate Singular value decomposition of A. 0 1 1 1 0 1

T  Because AA0 1 1 , then the eigenvalue of AAT are 3,   1,   0 .  1 2 3  1 1 2 1   1   1        The corresponding eigenvectors are  1 ,  1 ,    1 , 1  2  3         2 Only.  0   1ILLEGAL.  111 632   is 123 1 1 1 V  ,,    (38)    6 2 3 1 2 3  210  63file Then 3Version 0 0 2  TT  0 30 VAAV 0 1 0 , where  . (39)   this  00 01 000

by 1 1 6 2 1 0 11 1 0 1 22 UAV  1   11 3  Online1   (40) 6211 0 1 1 00  22 2 0 6 So the Singular valuemade decomposition of A is 1  1 2 11 6 6 6  T 22 3 0 0 11 AUOV ,0   . (41)   22 11 0 1 0 221 11  Book 333 Non-negative Matrix Factorization: For an arbitrarily given non negative matrix A, there are a non-negative matrix U and a non-negative matrix V, such that A=UV. That is to say that A non-negative matrix can be factorized into the product of two non- negative matrices.

Non-negative matrix factorization algorithm provides a new idea for matrix decomposition. Non-negative matrix factorization algorithm in the analysis and processing of images , text clustering and data mining , speech processing, robot control , biomedical engineering and chemical engineering has important application , in addition , Non-negative matrix factorization algorithm in the data signal analysis and complex object recognition have a very good application.

4. Conclusion

Firstly, some property of the transpose matrix or adjoint matrix A * of A may depend on the original matrix A. Such as the rank, determinant, especially adjoint matrix of A is linear combination of power of A. Secondly , this paper simplified formula about three kinds of equivalence relations between matrices of the same type , such as A , B be matrices of the same type , A=PBQ, where P , Q are invertible matrices . Thirdly, it discussed the relations between matrices of different types with the operation of the matrix. And using addition and multiplication operation of matrices to establish relationship between general matrix and symmetric matrix (skew-symmetric matrix), upper (lower) triangular matrix, invertible matrix, nilpotent matrix. Only.ILLEGAL. Acknowledgment This work was supported in part by the Education Departmentis of Heilongjiang Province science and technology research projects 12533068.

References file [1] J. Zhi, “Relationship of matrices equivalence, similarity and congruence”, Mudanjiang Normal University, no. 3, (2011). Version [2] TONGJI University Department of Mathematics, linear algebra, Beijing: Higher Education Press, (2007). [3] H. X. Liu, “Comparison Theorems in Linear Algebrathis”, Beijing: China Machine Press , (2009). [4] S. W. Bai, “Comparison Theorems in Linear Algebra”, Harbin : Heilongjiang Education Press , (1996). [5] A. L. He, Y. M. Ma, H. Liu and Y. H. Chen, “A note on matrix similarity”, J. of Shandong institute of light industry, vol. 19, no. 3, (2005). by [6] Q. H. XIE, “Applications of rank to the similarity of matrices”, Studies in college mathematics, vol. 15, no. 1, (2012). Online [7] Y. C. Bu and L. H. Ling, “Application of matrix in matrix theory in Teaching”, J. of Higher Correspondence Education (Natural Science), vol. 25, no. 4, (2012). [8] S. J. Duan and L. P. Sun, “Representation of Fibonacci sequence of numbers via matrix and determinants”, J. of Zhengzhou University of Light Industry (Natural Science), vol. 25, no. 5, (2010), pp. 117-119. [9] D. Serre, “Matricesmade Theory and Applications, Translated from Les Matrices: The´ orie et pratique”, published by Dunod (Paris), (2002); Springer-Verlag New York, Inc. [10] J. Hefferon, “Linear Algebra”, Saint Michael's College Colchester , Vermont USA [email protected] , (2000), pp. 347-412

Book Authors Li Fengxia, she received her bachelor's degree of Science in Harbin Normal University, (2004) and master's degree of Science in Harbin Normal University, (2007) now she works in Department of Information Science of Heilongjiang International University, Harbin. Her major in the fields of study are Algebra of Basic mathematics.

Only.ILLEGAL. is file Version this by Online made

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