DoishJournls 2041-1189 Doish Journl of Eductionl Reserch nd Reviews Vol 2(1) pp 001-007 Jnury, 2015 http://wwwdoishjournlsorg/djerr Copyright 2015 Doish Journls Originl Reserch Article Algebr of Mtrices nd Determits with MAXIMA Entrnce Niyzi Ari nd Svş Tuylu* Computer Science, Nigerin Turkish Nile University, Nigeri Accepted 24th December, 2014 The methods of clculus lie t the hert of the physicl sciences nd engineering Mxim cn help you mke fster progress, if you re just lerning clculus The exmples in this reserch pper will offer n opportunity to see some Mxim tools in the context of simple exmples, but you will likely be thinking bout much hrder problems you wnt to solve s you see these tools used here This reserch pper includes liner lgebr with MAXIMA Keywords: Algebr, Mtrices, Determits nd Mxim 1 INTRODUCTION Mxim is system for the mnipultion of symbolic nd numericl expressions, including differentition, integrtion, Tylor series, Lplce trnsforms, ordinry differentil equtions, systems of liner equtions, polynomils, sets, lists, vectors, mtrices, tensors, nd more Mxim yields high precision numeric results by using exct frctions, rbitrry precision integers, nd vrible precision floting point numbers Mxim cn plot functions nd dt in two nd three dimensions Mxim source code cn be compiled on mny computer operting systems, including Windows, Linux, nd McOS X The source code for ll systems nd precompiled binries for Windows nd Linux re vilble t the SourceForge file mnger Mxim is descendnt of Mcsym, the legendry computer lgebr system developed in the lte 1960s t the Msschusetts Institute of Technology It is the only system bsed on tht effort still publicly vilble nd with n ctive user community, thnks to its open source nture Mcsym ws revolutionry in its dy, nd mny lter systems, such s Mple nd Mthemtic, were inspired by it The Mxim brnch of Mcsym ws mintined by Willim Schelter from 1982 until he pssed wy in 2001 In 1998 he obtined permission to relese the source code under the GNU Generl Public License (GPL) It ws his efforts nd skills tht mde the survivl of Mxim possible MAXIMA hs been constntly updted nd used by resercher nd engineers s well s by students 2 ALGEBRA OF MATRICES 21 Clssicl Method 211 Introduction A mtrix A is rectngulr rry of sclrs usully presented in the following form: m1 mn (21) Corresponding Author: svstyludf@hotmilcom
N i y z i A r i n d S v ş T u y l u D o n n J E d u R e s R e v 002 The rows of the mtrix A re m horizontl lists of sclrs, nd the columns re n verticl lists of sclrs A squre mtrix is mtrix with the sme number of rows s columns An n _ n squre mtrix is sid to be of order n (nsqure mtrix): n1 (22) A digonl mtrix consists of the elements with the sme subscripts: 11 (23) The tce of A is the sum of the digonl elements: tr ( ) 11 33 A (24) A unit mtrix I is n-squre mtrix with 1 s on the digonl nd 0 s elsewhere write the rows of A s the columns of A T write the columns of A s the rows of A T Formlly, the ith row, jth column element of A T is the jth row, ith column element of A: T A A (29) ji If A is n m * n mtrix then A T is n n * m mtrix 11 21 m1 T A 12 (210) 1n 212 Mtrix Addition Consider the two mtrices A nd B b with the sme size m*n mtrices The sum of these mtrices is obtined by dding corresponding elements from these mtrices 11 b11 12 b12 1 n b1 n A B 21 b21 b (211) m 1 bm 1 m2 bm 2 mn b mn 1 0 1 0 1 (25) 2121 Exmples Exmple 21 Given two mtrices s 0 0 (26) A squre mtrix is upper tringulr if ll entries below the min digonl re equl to 0: A mtrix is symmetric if symmetric elements, mirror elements with respect to the digonl re equl, (27) ji The inverse of squre mtrix A, sometimes clled reciprocl mtrix, is mtrix A -1 such tht -1 A* A = I (28), where A is the identity mtrix A T creted by ny one of the following equivlent ctions: reflect A over its min digonl (which runs top-left to bottom-right) to obtin A T b b A 11 12 11 12 nd 21 B b21 b A B b b 11 11 12 12 21 b21 b Exmple 22 Given two mtrices s A 1 2 4 5 nd 3 4 B 6 7 A 14 2 5 5 7 B 36 4 7 9 11 nd 1 4 2 5 3 3 A B 3 6 4 7 3 3 wwwdoishjournlsorg
N i y z i A r i n d S v ş T u y l u D o n n J E d u R e s R e v 003 213 Sclr Multipliction The product of the mtrix A by sclr k is the mtrix obtined by multiplying ech element of A by k k11 k12 k1 n ka k21 k (212) km 1 km2 k mn 2131 Exmples Exmple 23 Given the mtrix A s 11 12 11 12 A nd k=c 21 Exmple 24 Given the mtrix A s 1 2 3 4 c c ka c21 c A nd k=2 2 4 ka 6 8 214 Mtrix Multipliction Consider the mtrices A B which re ik nd bkj mtrices such tht the number of columns of A is equl to the number of rows of B Given the mtrix A s b b A 11 12 11 12 nd 21 B b21 b AB x b b 11 12 11 12 x 21 b21 b ( b b ) ( b b ) ( 21b11 b21 ) ( 21b12 b ) 11 11 12 21 11 12 12 AB x Exmple 26 Given the mtrix A s A 4 3 8 7 nd 2 1 B 6 5 4 3 8 7 AB x x, 2 1 6 5 C (4x8 3x6) (4x7 3x5) 50 43 A x B (2x8 2x6) (2x7 1x5) 28 19 22 MAXIMA APPLICATIONS 221 Introduction 222 Mtrix Addition Is A n m*n mtrix nd B is p*n mtrix, then the product AB is the m*n mtrix whose -entry is obtined by multiplying the ith row of A by the jth column of B 11 ip b11 b1 j b1 n i1 ip x C = AxB = bp1 bpj b pm m 1 mp c11 c1 n C = AxB c (213) cm 1 c mn where 223 Sclr Multipliction c b b b b (214) i1 1 j i2 2 j ip pj ik kj k1 2141 Exmples Exmple 25 p wwwdoishjournlsorg
N i y z i A r i n d S v ş T u y l u D o n n J E d u R e s R e v 004 Tble 1: Some selected mtrix functions from MAXIMA re given in the following tble: Function entermtrix(m,n) genmtrix() mtrix(row1,rown) invert(m) trnspose(m)l tringulrize(m) zeromtrix(m,n) ddcol(a,b) ddrowl(a,b) row(a,i) col(a,j) Description Returns n m by n mtrix reding the elements Returns mtrix generted from Returns rectngulr mtrix Returns the inverse of mtrix M Returns the trnspose of mtrix M Returns the upper tringulre of mtrix M Returns n m by n mtrix, ll elements of zero Mtrices cn be stcked Mtrices cn be stcked Returns the ith row Returns the jth column 224 Mtrix Multipliction Exmple 2 225 Numericl Exmple Exmple 3 Exmple 1 Exmple 4 wwwdoishjournlsorg
N i y z i A r i n d S v ş T u y l u D o n n J E d u R e s R e v 005 The th cofctor C of n n x n mtrix A is (-1) i+j det M The formul (-1) i+j ssigns +1 or -1 to det M depending on whether i + j is even or odd Using the cofctors, the det of A is defined For ny (n) x (n) mtrix Exmple 5 A n1 (35) The determit of A is defined s: det A A (sgn ) (36) where j j j 1 2 n 1j1 2j1 nj n Exmple 31 Consider the following mtrix nd clculte its determit using cofctors: 3 DETERMİNANTS 31 Clssicl Methd 311 Definitions The determit det A of n nxn mtrix A, for n 2, is defined s the Lplce expnsion of the determit, using submtrices of A For ny 2 x 2 mtrix A n1 Solution: By definition, det A C C C (37) The cofctors 11 11 12 12 13 13 C re: A 11 12 21 (31) C ( 1) det ( ) 11 23 11 33 23 32 32 33 is the det A defined s: det A (32) 11 12 21 For ny 3x3 mtrix A 11 12 13 21 23 31 32 33 is the det A defined s: (33) det A (34) 11 33 11 23 32 12 21 33 12 23 31 13 21 32 13 31 The th minor M of (n) x (n) mtrix A is the (n - 1) x(n - 1) submtrix of A obtined by deleting the ith row nd the jth column of A Assuming tht the determits det M of the minors of A re known, nd use them to define the determit of A itself C C ( 1) det ( ) 1 2 21 23 12 21 33 23 31 31 33 ( 1) det ( ) 1 3 21 13 21 32 31 31 32 Therefore, det A C C C nd 11 11 12 12 13 13 det A ( ) ( ) ( ) 11 33 23 32 12 21 33 23 31 13 21 32 31 det A 11 33 11 2332 12 2133 12 2331 13 2132 13 31 312 Crmer s Rule The Crmer s rule is procedure for solving liner systems, using the determits Consider the liner system wwwdoishjournlsorg
N i y z i A r i n d S v ş T u y l u D o n n J E d u R e s R e v 006 Ax b, where the Ax is the coefficient mtrix, x is the unknown vector, nd b is the column vector The Crmer s rule is defined for ech 1i n 32 MAXIMA APPLICATIONS 321 Introduction Determit(M): returns the determit of the mtrix M Exmple 33 by det Α(i/ b) xi, (38) det Α where the numertor is the mtrix obtined from Ax by replcing the ith column of Ax by b Exmple 32 Exmple 34 Given the following liner system x y 3z 10 x 2y z 5 x y z 5 Its mtrix form is Exmple 35 Ax = b 1 1 3 x 10 1 2 1 y 5 1 1 1 z 5 Solution: Exmple 36 det A 11 33 11 23 32 12 21 33 12 23 31 13 21 32 13 31 1 1 3 A 1 2 1 1 1 1 det A 2 10 1 3 A(1/ b) 5 2 1, 5 1 1 det( A(1/ b)) 15 1 10 3 A(2 / b) 1 5 1, 1 5 1 det( A(2 / b)) 10 1 1 10 A(3 / b) 1 2 5 1 1 5 det( A(3 / b)) 5 det A(1/ b) 15 det A(2 / b) 10 det A(3 / b) 5 x 75, y 5, z 25 det A 2 det A 2 det A 2 wwwdoishjournlsorg
N i y z i A r i n d S v ş T u y l u D o n n J E d u R e s R e v 007 4 CONCLUSION The reserch pper cn pply ech nd every prt of Algebr of Mtrices nd Determits, help ppliction of the physicl sciences nd engineering, mke fster progress, nd help to understnd Algebr of Mtrices nd Determits fster The pper prticulrity helps to understnd prts of Liner Algebr nd is going to extend to other prts of the Liner Algebr 5 ACKNOWLEDGEMENTS I would like to thnk the Mxim developers nd Nigerin Turkish Nile University for their friendly help REFERENCES [1] R H Rnd, Introduction to Mxim, [2] R Dodier, Miniml Mxim, 2005 [3] https://wwwmutexsedu/mxim/mxim\_19html [4] N Ari, G Apydin, Symbolic Computtion Techniques for Electromgnetics with MAXIMA nd MAPLE, Lmbert Acdemic Publishing, 2009 [5] Fred Szbo Liner Algebr, An Introduction Using MATHEMATICA Acdemic Press 2000 [6] S Lipschutz, M Lipson, Liner Algebr Schum s Outlines McGrw Hill2009 [7] S Lipschutz, Liner Algebr Schum s Solved Problems Series McGrw Hill 1989 [8] Minh Vn Nguyen, Liner Algebr with MAXIMA http://wwwmthsuweduu/students/undergrdute/m S/AfterMth [9] N ARI, Engineering Mthemtics with MAXIMA SDU, Almty, 2012 [10] Niyzi ARI, Lecture notes, University of Technology, Zurich, Switzerlnd [11] Niyzi ARI, Symbolic computtion of electromgnetics with Mxim (2013) [12] http://mximsourceforgenet/ wwwdoishjournlsorg