Introduction to the General operation of Matrices

Transcription

1 Introduction to the General operation of Matrices CLAUDE ZIAD BAYEH 1, 1 Faculty of Engineering II, Lebanese University EGRDI transaction on mathematics (004) LEBANON claude_bayeh_cbegrdi@hotmail.com NIKOS E.MASTORAKIS WSEAS (Research and Development Department) Agiou Ioannou Theologou , Zografou, Athens,GREECE mastor@wseas.org Abstract: - The General operation of Matrices is an original study introduced by the author in the mathematical domain. The basic idea of the General operation of Matrices is similar to the traditional operation of matrices, but moreover the General operation of Matrices is more general and contains many forms and ways of multiplying, adding, subtracting and dividing many matrices. The main goal of introducing the General operation of Matrices is to facilitate the writing of many complicated equations in simple matrices. In this paper a brief study is introduced with the definition of the General operation of Matrices and simple applications are developed in order to give idea about the importance of this new concept of operation between Matrices. Many studies will follow this one in order to find more applications in mathematics and all scientific domains. Key-words:- General operation of Matrices, Matrix, Addition, Multiplication, Subtracting, Dividing, Relation between matrices. 1 Introduction In mathematics, a matrix (plural matrices) is a rectangular array of numbers, symbols, or expressions [1-]. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements is Matrices of the same size can be added or subtracted element by element [3-4]. The rule for matrix multiplication is more complicated, and two matrices can be multiplied only when the number of columns in the first equals the number of rows in the second [5-7]. A major application of matrices is to represent linear transformations, that is, generalizations of linear functions such as f(x) 4x. For example, the rotation of vectors in three dimensional space is a linear transformation [8-9]. The product of two matrices is a matrix that represents the composition of two linear transformations. Another application of matrices is in the solution of a system of linear equations. If the matrix is square, it is possible to deduce some of its properties by computing its determinant. For example, a square matrix has an inverse if and only if its determinant is not zero. Eigenvalues and eigenvectors provide insight into the geometry of linear transformations. Matrices find applications in most scientific fields. In physics, matrices are used to study electrical circuits, optics, and quantum mechanics [10-11]. In this paper, the author introduced the General operation of Matrices which is the general case of the traditional operation between matrices. The concept of manipulating the General operation of Matrices is dierent from the traditional operation between matrices. The general operation of matrices is introduced in order to facilitate the writing of many complicated expressions into matrices and to facilitate the operation between matrices as we want. We can operate the Columns of the first matrix by the columns of the second matrix, or the columns of the first matrix by the Lines (rows) of the second matrix, or Lines of the first matrix by the line of the second matrix, or Lines of the first matrix by the columns of the second matrix. This is not the case of the traditional operation between matrices. Mainly we have two operations between matrices, the first operation is between the elements of the two matrices and the second is the operation between the operated elements. For example: (aa ( 1 )bb) is the first operation between two elements of the two matrices, with ( 1 ) can be replaced by an operator (, -, *, ), the second operator ( ) is the operator ISBN:

2 between the operated elements such as (aa 1 bb) (cc 1 dd). ( ) can be replaced by an operator (, -, *, ). Many studies will follow this paper in order to find other applications in many domains as in science, engineering and mathematics. In the second section, a definition of the general operation of matrices is presented. in the section 3, practical examples of the General operation of matrices are presented. and finally, a conclusion is presented in the section 4. Definition of the General operation of matrices The general operation of matrices is denoted by: MM 1 MM 1 (1) MM 1 MM is the form of operation between two matrices for example it is written as following: - means the operation between the Columns of the first matrix and the Columns of the second matrix. - means the operation between the Columns of the first matrix and the Lines of the second matrix. - means the operation between the Lines of the first matrix and the Lines of the second matrix. - means the operation between the Lines of the first matrix and the Columns of the second matrix. 1, is the first operation between the elements of the first and second matrices. It can be replaced by (,, oooo )., is the secondary operation between the elements of the first and second matrices. It can be replaced by (,, oooo ). Examples: AA aa bb and BB ee cc dd AA 1 BB aa bb cc dd 1 ee (aa 1 ee) (cc 1 gg) (bb 1 ee) (dd 1 gg) (aa 1 ) (cc 1 h) (bb 1 ) (dd 1 h) AA 1 BB aa bb cc dd 1 ee (aa 1ee) (cc 1 ) (bb 1 ee) (dd 1 ) (aa 1 gg) (cc 1 h) (bb 1 gg) (dd 1 h) AA 1 BB aa cc bb dd 1 ee gg h (aa 1ee) (bb 1 ) (aa 1 gg) (bb 1 h) (cc 1 ee) (dd 1 ) (cc 1 gg) (dd 1 h) AA 1 BB aa cc bb dd 1 ee gg h (aa 1ee) (bb 1 gg) (aa 1 ) (bb 1 h) (cc 1 ee) (dd 1 gg) (cc 1 ) (dd 1 h).1 Examples of the general operation of matrices In this section, we give some examples in order to understand the principle of the general operation of matrices and how it works. It is very important to understand the basis of operation in order to understand the whole operations. AA aa bb and BB ee cc dd.1.1 The first operation is * and the second is - AA BB aa bb (aa ee) (cc gg) (bb ee) (dd gg) (aa ) (cc h) (bb ) (dd h) AA BB aa bb (aa ee) (cc ) (bb ee) (dd ) (aa gg) (cc h) (bb gg) (dd h) AA BB aa bb (aa ee) (bb ) (aa gg) (bb h) (cc ee) (dd ) (cc gg) (dd h) ISBN:

3 AA BB aa bb (aa ee) (bb gg) (aa ) (bb h) (cc ee) (dd gg) (cc ) (dd h).1. The first operation is * and the second is AA BB aa bb (aa ee) (cc gg) (bb ee) (dd gg) (aa ) (cc h) (bb ) (dd h) AA BB aa bb (aa ee) (cc ) (bb ee) (dd ) (aa gg) (cc h) (bb gg) (dd h) AA BB aa bb (aa ee) (bb ) (aa gg) (bb h) (cc ee) (dd ) (cc gg) (dd h) AA BB aa bb (aa ee) (bb gg) (aa ) (bb h) (cc ee) (dd gg) (cc ) (dd h) This equation is similar to the traditional operation between two matrices which gives the same result as following AA BB aa bb ee cc dd (aa ee) (bb gg) (aa ) (bb h) (cc ee) (dd gg) (cc ) (dd h).1.3 The first operation is and the second is AA BB aa bb cc dd ee (aa ee)(cc gg) (bb ee)(dd gg) (aa )(cc h) (bb )(dd h) AA BB aa bb cc dd ee (aa ee)(cc ) (bb ee)(dd ) (aa gg)(cc h) (bb gg)(dd h) AA BB aa bb cc dd ee (aa ee)(bb ) (aa gg)(bb h) (cc ee)(dd ) (cc gg)(dd h) AA BB aa bb cc dd ee (aa ee)(bb gg) (aa )(bb h) (cc ee)(dd gg) (cc )(dd h).1.4 The first operation is * and the second is - with two rectangular matrices (3 columns and lines) aa bb cc h ii AA and BB gg dd ee jj kk ll aa bb cc AA BB dd ee gg h ii jj kk ll (aa gg) (dd jj) (bb gg) (ee jj) (cc gg) ( jj) (aa h) (dd kk) (bb h) (ee kk) (cc h) ( kk) (aa ii) (dd ll) (bb ii) (ee ll) (cc ii) ( ll) aa bb cc AA BB dd ee gg h ii jj kk ll lines in the first matrix must be equal to the number of columns in the second matrix. AA BB aa bb (aa gg) (bb h) (cc ii) (aa jj) (bb kk) (cc ll) (dd gg) (ee h) ( ii) (dd jj) (ee kk) ( ll) AA BB aa bb columns in the first matrix must be equal to the number of lines in the second matrix..1.5 The first operation is * and the second is - with two rectangular matrices ( columns and 3 lines) aa bb gg h AA cc dd and BB ii jj ee kk ll ISBN:

4 aa bb gg h AA BB cc ee kk ll (aa gg) (cc ii) (ee kk) (bb gg) (dd ii) ( kk) (aa h) (cc jj) (ee ll) (bb h) (dd jj) ( ll) aa bb gg h AA BB cc ee kk ll lines in the first matrix must be equal to the number of columns in the second matrix. aa bb gg h AA BB cc ee kk ll (aa gg) (bb h) (aa ii) (bb jj) (aa kk) (bb ll) (cc gg) (dd h) (cc ii) (dd jj) (ee kk) (dd ll) (ee gg) ( h) (ee ii) ( jj) (ee kk) ( ll) aa bb gg h AA BB cc ee kk ll columns in the first matrix must be equal to the number of lines in the second matrix.. General Case of operation between matrices 1 BB jj ll ll MM ii () -jj is the number of lines for the two matrices. The number of lines for the two matrices should be equal when the operation between the two matrices is columns of the first matrix by the columns of the second matrix. - ii is the number of columns for the first matrix AA. -ll is the number of columns for the second matrix BB. The result matrix has the number of lines equal to ll and the number of columns equal to ii. AA 1 1 BB 3 MM 3 1 [ ] 1 [ ] 1 BB ll ll jj MM ii (3) -jj is the number of lines for the first matrix and it is equal to the number of columns for the second matrix. The number of lines for the first matrix must be equal to the number of columns in the second matrix when the operation between the two matrices is columns of the first matrix by the lines of the second matrix. - ii is the number of columns for the first matrix AA. -ll is the number of columns for the second matrix BB. The result matrix has the number of lines equal to ll and the number of columns equal to ii. AA 1 1 BB 3 1 [ ] 1 3 MM 1 BB ll jj ii MM ll (4) -jj is the number of lines for the first matrix. - ii is the number of columns for the first matrix AA. The number of columns for the first matrix must be equal to the number of columns in the second matrix when the operation between the two matrices is Lines of the first matrix by the lines of the second matrix. -ll is the number of lines for the second matrix BB. The result matrix has the number of lines equal to jj and the number of columns equal to ll. AA BB 3 [ ] 1 [ ] 1 MM 1 BB ii jj ll MM ll (5) -jj is the number of lines for the first matrix. -ii is the number of columns for the first matrix AA. The number of columns for the first matrix must be equal to the number of lines in the second matrix when the operation between the two matrices is Lines of the first matrix by the columns of the second matrix. -ll is the number of columns for the second matrix BB. ISBN:

5 The result matrix has the number of lines equal to jj and the number of columns equal to ll. AA BB 3 1 [ ] OOpp 3 MM 3.3 The Case when we have only the first operator 11 When we have only one operator we consider it as the first operator. In this case the two matrices must have the same dimensions for example AA ii jj and BB ii jj with jj is the number of lines and ii is the number of columns for both matrices. The operation will be only between the elements of two matrices which are located in the same position. AA aa bb and BB ee cc dd AA( 1 )BB aa bb cc dd ( 1 ) ee AA( 1 )BB aa 1ee bb 1 cc 1 gg dd 1 h For example: consider that the first Operation ( 1 ) is replaced respectively by (, *, -, ). Therefore the operation of two matrices will be as following: AA aa bb cc dd and BB ee thus gg h aa ee bb AA()BB cc gg dd h aa ee bb AA( )BB cc gg dd h aa ee bb AA()BB cc gg dd h AA()BB aaee bb ccgg ddh Remark: The operation between two matrices AA( )BB is not the same as the traditional operation AA BB. 3 Practical examples of the General operation of matrices In this section we are going to see some practical examples in mathematics in order to give an idea about the importance of this new method of operation between matrices. Example 1 Suppose we have three polynomial equations of the second order as following: yy 1 aaxx bbbb cc yy ddxx eeee yy 3 ggxx hxx ii yy 1 aa bb cc xx yy dd ee xx yy 3 gg h ii 1 It can be written also in the following form aa dd gg xx [yy 1 yy yy 3 ] bb ee h xx cc ii 1 Example Suppose we have four polynomial equations as following: yy 1 xx (xx1) yy xx(1 1 ) xx yy 3 xx yy 4 xx(4xx 3) yy 1 1 yy xx xx yy 3 yy 4 xx xx ( ) xx xx1 1 1 xx 4xx 3 Example 3 Suppose we have four polynomial equations as following: yy 1 (xx xx)(xx 3) yy xx 1 (xx ) xx yy 3 (xx 3 xx)( 3) yy 4 xx 3 1 ( ) xx yy 1 yy yy 3 yy xx xx 4 xx 3 xx 3 1 xx ISBN:

6 Example 4 Suppose we have three polynomial equations of the second order as following: yy 1 aaxx 3 bbxx cc It can be written in the following form aa xx 3 [yy 1 ] bb xx cc 1 It can be written also in the following form aa [yy 1 ] bb [xx 3 xx 1] cc Example 5 Suppose we have four polynomial equations as following: yy 1 xx xx xx3 yy xx 1 xx xx yy 3 xx 3 xx 3 yy 4 xx 3 1 xx yy 1 yy yy 3 yy xx xx 4 xx 3 xx 3 1 xx 4 Conclusion In this paper, the author introduced a new and original way of operation between matrices. The traditional operation between matrices is just a part of the general operation of matrices. The general operation of matrices is introduced in order to facilitate the writing of many complicated expressions into matrices and to facilitate the operation between matrices as we want. We can operate the Columns of the first matrix by the columns of the second matrix, or the columns of the first matrix by the Lines (rows) of the second matrix, or Lines of the first matrix by the line of the second matrix, or the Lines of the first matrix by the columns of the second matrix. This is not the case of the traditional operation between matrices. Mainly we have two operations between matrices, the first operation is between the elements of the two matrices and the second is the operation between the operated elements. For example: (aa ( 1 )bb) is the first operation between two elements of the two matrices, with ( 1 ) can be replaced by an operator (, -, *, ), the second operator ( ) is the operator between the operated elements such as (aa 1 bb) (cc 1 dd). ( ) can be replaced by an operator (, -, *, ). Many studies will follow this paper in order to find other applications in many domains as in science, engineering and mathematics. References: [1] Arnold Vladimir I.; Cooke Roger, Ordinary dierential equations, Berlin, DE; New York, NY:Springer-Verlag, ISBN (199). [] Artin Michael, Algebra, Prentice Hall, ISBN , (1991). [3] Association for Computing Machinery, Computer Graphics, Tata McGraw Hill, ISBN , (1979). [4] Baker Andrew J., Matrix Groups: An Introduction to Lie Group Theory, Berlin, DE; New York, NY: Springer-Verlag, ISBN , (003). [5] Bau III David, Trefethen Lloyd N., Numerical linear algebra, Philadelphia, PA: Society for Industrial and Applied Mathematics, ISBN , (1997). [6] Bretscher Otto, Linear Algebra with Applications (3rd ed.), Prentice Hall, (005). [7] Bronson Richard, Schaum's outline of theory and problems of matrix operations, New York: McGraw Hill, ISBN , (1989). [8] Brown William A., Matrices and vector spaces, New York, NY: M. Dekker, ISBN , (1991). [9] Coburn Nathaniel, Vector and tensor analysis, New York, NY: Macmillan, OCLC 10988, (1955). [10] Conrey J. Brian, Ranks of elliptic curves and random matrix theory, Cambridge University Press, ISBN , (007). [11] Gilbarg David, Trudinger Neil S., Elliptic partial dierential equations of second order (nd ed.), Berlin, DE; New York, NY: Springer- Verlag, ISBN , (001). ISBN: