Character analysis on linear elementary algebra with max-plus operation

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1 Available online at WSN 100 (2018) EISSN Character analysis on linear eleentary algebra with ax-plus operation ABSTRACT Kalfin 1, Jufra 2, Nora Muhtar 2, Subiyanto 3, Sudradjat Supian 4, * 1 Master Progra in Matheatics, Faculty of Matheatics and Natural Sciences, Universitas Padjadjaran, Indonesia 2 Departent of Matheatics, Faculty of Matheatics and Natural Sciences, Universitas Halu Oleo, Indonesia 3 Departent of Marine Science, Faculty of Fishery and Marine Science, Universitas Padjadjaran, Indonesia 4 Departent of Matheatics, Faculty of Matheatics and Natural Science, Universitas Padjadjaran, Indonesia *E-ail address: sudradjat@unpad.ac.id This paper discusses the further characteristic that exists in ax-plus algebra by coparing the characteristics that exist in ordinary linear algebra. The steps taken are to exaine what happens to linear algebra, then applied to ax-plus algebra with respect to the existing operations. Characteristics of atrix operations and differences that occur in linear algebra and are et in ax-plus algebra with ax and plus operations. However, ax-plus algebra does not satisfy the vector space characteristics. Keywords: Max-Plus Algebra, atrix, vector space, eigenvalues and eigenvectors 1. INTRODUCTION Max-Plus Algebra is one part of basic science in the field of atheatics, especially algebra. Max-plus algebra is able to decipher a particular type, fro a nonlinear syste in linear algebra to a linear syste in ax-plus algebra 3. Algebra Max-Plus was first ( Received 12 May 2018; Accepted 25 May 2018; Date of Publication 25 May 2018 )

2 introduced by Baccelli, Cohen, Olsder, and Quadrat, in 1992, and continues to be developed to date. The ax-plus algebra is the set { } with R the set of all real nubers fitted with ax and plus operations 11,14,19. The basic operation of ax-plus algebra is ax (axiu) denoted by the sybol and plus (plus) is denoted by the sybol, with the two operations for each 1,11,28. Multiplication of two atrices in ax-plus algebra has a siilarity to the ultiplication of two atrices in ordinary algebra. Multiplication of two atrices in ordinary algebra using operations (ultiplication) and (plus), while ultiplication of two atrices in ax-plus algebra, using (plus) and (axiu) operations respectively instead of operations with and with + 23,25,27,29. In axplus algebra, single exponential atrix values show that for single ax-plus values it gives a sequence of agnitude approaches of single values of independent classical atrix paraeters 12. Max-plus operation has been widely practiced by previous researchers. For exaple Iaev and Judd 8 deterine eigenvector coputation on an inverted atrix in ax-plus algebra. Butkovic and Maccaig 4 deterine the eigenvectors of integers and eigenvectors in ax-plus algebra. Lu et.al 20 analyzed the reachability of tied autoata using ax-plus algebra. Model and algebraically analyze network probles, such as scheduling probles in a project can be seen in 2. Krivulin 18 discusses the dynaics odel of the fork-join queue network with the buffer capacity of the finite into a ax-plus algebra atrix equation. The deterination of eigenvalues and eigenvectors has a role in optiization solutions such as project anageent and scheduling can be seen in 9. The object in the analysis is to copare the characters in the regular Linear Algebra with the ax-plus Algebra characters by observing the existing operations. The goal is to get the character difference found on a regular Linear Algebra with Algebra ax-plus. 2. MATERIALS AND METHODS Max-plus Algebra Definition 1: Given the set { } denoted by, for R the set of all real nubers with axiu operation, denoted by and the addition operation denoted by. if given any nuber of x and y, then x y is the axiu value of one of these nubers, and x y is the su of the two nubers. Further ( { } ) is denoted by and denoted by ε. The ε eleent is a neutral eleent to the operation and is the eleent of identity against operation, and is defined: For each Matrix and Vector on Max-plus Algebra Matrices and vectors have a close connection in algebra, and then in this section will be discussed about the atrices and vectors in -111-

3 Matrix The atrix set for in is denoted by. In the atrix, denotes the nuber of rows and denotes the nuber of coluns. In general, the atrix is written as follows: Matrix A for the input value of row and colun denoted by The su and axiu on the atrix and vector Max-Plus Algebra is defined in a different way axiu and plus. Definition 2: a. For the axiu is defined A B by: b. The transpose of the atrix is denoted by and specifically in Max-plus Algebra is defined c. Max-plus Algebra's identity atrix En is defined as follows: { d. For the square atrix and k positive integers, the power on A is denoted by is defined: For e. For any atrix and any scalar is defined as follows: 8. Operations and in can be extended to atrix operations as in the following definitions: Definition 3: Given { } a. For defined is a atrix whose eleents are ij: -112-

4 ( ) For i = 1, 2,..., and j = 1,2,., n then is a atrix whose eleents are ij:( ) For i = 1,2,.., and j = 1, 2,, n b. For, defined is a atrix whose eleents are ij: ( ) for i = 1,2,..., and j = 1,2,., n 22, Vector A eber of is called a Max-Plus vector if coponent of vector is denoted by or The colun to of the identity atrix is known as the colun vector to in This vector is denoted by In other words, is the enter to j in the vector 8, Eigenvalues and Eigenvectors on Max-Plus Algebra Definition 4: Given the atrix then the scalar is called the eigenvalue of the atrix A if there is a vector containing at least one eleent instead of ε such that Furtherore, is called the eigenvector of the atrix A 1,5,6, RESULT AND DISCUSSION In this section, we analyze the characteristics of Max-plus Algebra. The discussion includes: vector space, atrix characteristics and lifts characteristics in Max-plus Algebra. This discussion begins with the following vector space Vector Space In ordinary linear algebra, a vector satisfies the vector space if it satisfies the ten ost iportant properties of the vectors in based on the ten properties present in linear algebra will be proved whether the ax-plus algebra also satisfies the vector space based on the ten ost iportant properties of the vectors in taking into account the operations present in ax-plus algebra. If the following axios are satisfied by all on and by all scalars (nuber ) and, then is naed as vector space a. If and are objects in, then is in b. c. d. There is a in so for all in e. For every in, there is in which call negative so -113-

5 f. If is any scalar and is any object in, then in. g. ( ) ( ) h. ( ) ( ) i. ( ) ( ) j. Evidence: Take and a. b. c. ( ) d. so ( ( ) ( ) ( )) ( ( ) ( ) ( )) ( ) e. so -114-

6 In this section cannot be satisfied because the ax operation of its inverse eleent does not exist, as seen in the following theore. Theore 1. If the operation in ( { } ) is idepotent, then the inverse eleent to the operation does not exist. Contradictions with so the inverse eleent to the operation does not exist. f. Let k scalar for then ( ) g. ( ) ( ) (( ) ( ) ( ) ( ) ( ) ( )) ( ) ( ) ( ) ( ) h. ( ) ( ) ( ) ( ) ( ) ( ) i. ( ) ( ) ( ( ) ( ) ( )) (( ) ( ) ( ) ) ( ) ( ) j. ( ) = u -115-

7 Fro the above proof, it appears that does not eet the properties of the vector space. Because at ax operation, it has no inverse eleent so can not be fulfilled. In this case is not a vector space Characteristics Matrix Max-Plus Algebra In linear algebra, the operation of the atrix is known to soe of the properties found in the operation of the atrix. As with linear algebra, ax-plus algebra also has the sae characteristics in linear algebra with respect to the operations present in ax-plus algebra. The rules of arithetic that exist in linear algebra such as coutative law, associative law and distributive law also apply to ax-plus algebra. For ore details can be seen in the following discussion: Soe of the following properties apply to any atrix A, B, and C with corresponding sizes and undefined atrix operations, for loyal scalar Evidence: a. b. ( ) ( ) c. ( ) ( ) d. ( ) ( ) ( ) e. ( ) ( ) ( ) f. ( ) ( ) ( ) g. ( ) ( ) ( ) h. ( ) ( ) a. Take any atrix so ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) -116-

8 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) b. Take any atrix so ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) = ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) = ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) -117-

9 c. Take any atrix, and The row eleent i colun j atrix (A B) C is as follows: ( ) ( ) ( ) ( ) For and d. Take any atrix and. The row eleent i colun j atrix (A B) C is as follows: ( ( )) ( ) e. Take any atrix f. Take any atrix ( ) ( ) ( ) ( ) ( ), ( ) ( ) = = -118-

10 ( ) g. Take any atrix so ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) h. Take any atrix A so ( ) ( ) -119-

11 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) Characteristic of appointent Max-Plus Algebra Operating and in ax-plus algebra, operation is perfored first fro operation. Soe properties in the operation of the nubers on ax-plus algebra are as follows: a. b. c. d. Evidence: a. n n b

12 n n c. d. 4. CONCLUSION Based on the results of the research it can be concluded that the atrix of ax-plus algebra has the sae characteristics as the atrix in ordinary algebra with respect to the operations present in ax-plus algebra. The properties of vector space on ax-plus algebra do not satisfy the properties of vector spaces in ordinary linear algebra References 1 Akian, M. Bapat, R. & Gaubert, S., Asyptotics of the Perron eigenvalue and eigenvector using ax-algebra. Coptes Rendus de l'acadéie des Sciences - Series I Matheatics, 327(11) (1998) Baccelli, F., Cohen, G., Olsder, G. J. & Quadrat, J., Synchronization and Linearity: an algebra for discrete event syste. New York: Wiley-Interscience Butkovic, P. Max-algebra: the linear algebra of cobinatorics? Linear Algebra and its Applications 367 (2003) Butkovic, P. & Maccaig, M. On integer eigenvectors and subeigenvectors in the axplus algebra. Linear Algebra and its Applications, 438 (2013) Butkovic, P. Peruted ax-algebraic eigenvector proble is NP-coplete. Linear Algebra and its Applications, 428 (2008)

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