INTRODUCTION TO MATRIX ALGEBRA. a 11 a a 1n a 21 a a 2n...

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1 INTRODUCTION TO MATRIX ALGEBRA DEFINITIONOFAMATRIXANDAVECTOR DefiitioofamatrixAmatrixisarectagulararrayofumbersarrageditorowsad colums It is writte as a a 2 a a 2 a 22 a 2 () a m a m2 a m Theabovearrayiscalledamby(m )matrixsiceithasmrowsadcolumstypically upper-case letters are used to deote a matrix ad lower case letters with subscripts the elemets ThematrixAisalsooftedeoted Cosider the followig 3 3 example Ithismatrixa 3 =4ada 23 =3 A = a ij (2) (3) DefiitioofavectorAvectorisa-tupleofumbers ItwodimesioalspaceorR 2,a vectorwouldbeaorderedpairofumbers {x,y}ithreedimesioalspaceorr 3,avectoris a3-tuple,ie, {x,x 2,x 3 }SimilarlyforR Vectorsareusuallydeotedbylowercaseletterssuch asaorb,ormoreformally aor b 3 Row ad colum vectors 3RowvectorAmatrixwithoerowadcolums( )iscalledarowvectoritisusually writte x or x = ( x x 2 x 3 x ) Theuseoftheprime symbolidicateswearewritigthe-tuplehorizotallyasifitwerethe rowofamatrixnotethateachrowofamatrixisarowvectorarowvectormightbeasfollows (4) wherez 2 =4 z = ( 4 3 ) (5) Date: 4 March 2008

2 2 INTRODUCTION TO MATRIX ALGEBRA 32ColumvectorAmatrixwithoecolumadrows( )iscalledacolumvector Itiswritteas Acolumvectormightbeasfollows x = p = x x 2 x 3 x (6) 2 3 (7) NotethateachcolumofamatrixisacolumvectorItiscommotowritethecolumsofa matrixasa,a 2,a whereeachcolumvectora j isoflegthmasaexamplea 2 isgiveby Iequatio3,a 2 isgiveby a 2 = a 2 = a 2 a 22 a 32 a m2 (8) 3 4 (9) 2 VARIOUS TYPES OF MATRICES AND VECTORS 2SquarematricesAsquarematrixisamatrixwithaequalumberofrowsadcolums,ie m= 22TrasposeofamatrixThetrasposeofamatrixAisamatrixformedfromAbyiterchagig rowsadcolumssuchthatrowiofabecomescolumiofthetrasposedmatrixthetraspose isdeotedbya ora T ad 2 3 A = a ji whea = a ij (0) Ifa ij istheij thelemetofa,thea ij = a ji IfthematrixAisgiveby thea isgiveby A = () 5 2

3 INTRODUCTION TO MATRIX ALGEBRA 3 A = (2) SymmetricmatrixAsymmetricmatrixisasquarematrixAforwhich Aexampleofasymmetricmatrixis T = T = A = A (3) (4) IdetitymatrixTheidetitymatrixoforderwritteIorI,isasquarematrixhavigoes alog the mai diagoal(the diagoal ruig from upper left to lower right ad zeroes elsewhere) IfwewriteI= δ ij the δ ij = Thesymbol δ ij iscalledthekroeckerdelta {, i =j 0, i =j 25 Scalar matrix For ay scalar λ, the square matrix iscalledascalarmatrixaexampleis (5) (6) S = λ δ ij = λi (7) (8)

4 4 INTRODUCTION TO MATRIX ALGEBRA 26 Diagoal matrix A square matrix D = λ i δ ij (9) iscalledadiagoalmatrixnoticethat λ i varieswithiaexampleis (20) If a system of equatios i four variables was writte with this coefficiet matrix, we could solve the system by solvig each equatio idividually because each variable would appear i each equatio oly oce 27NullorzeromatrixTheullorzeromatrixisamatrixwitheachelemetbeigzero Itis deoted as = (2) Upper triagular matrix A matrix with all elemets below the mai diagoal equal to zero is called a upper triagular matrix a a 2 a 3 a 0 a 22 a 23 a a 33 a 3 A = (22) a m Specificallya ij =0ifi >jaslogasi <madj< 29 Lower triagular matrix A matrix with all elemets above the mai diagoal equal to zero is called a lower triagular matrix a a 2 a a 3 a 32 a 33 0 A = (23) a m a m2 a m3 a m Specificallya ij =0ifi <jaslogasi <madj< The followig two matrices are upper triagular ad lower triagular respectively

5 INTRODUCTION TO MATRIX ALGEBRA (24) 3 Sigle sums 3 Defiitio of a sigle sum 3 A NOTE ON SUMMATION NOTATION a i = a m +a m+ +a m+2 + +a (25) i=m For example, suppose we have a vector with the followig elemets a = ( a a 2 a 3 a 4 a 9 ) The 32 Properties of a sigle sum 32 Double sums 32 Defiitio of a double sum i= = ( ) (26) 6 a i = = 5 (27) i=3 i= ka i =k a i i= k =k +k+k+ +k = k i= (a i +b i ) = i= m a ij = j= i= m a j + j= a i + b i i= m j= a 2j + + m j= a j (28) =a +a 2 +a 3 + +a m +a 2 +a 22 +a a 2m (29) Cosider the followig matrix +a +a 2 +a 3 + +a m

6 6 INTRODUCTION TO MATRIX ALGEBRA A = Wecompute 4 i=2 3 j=2 a ijasfollows S =a 22 +a 23 +a 32 +a 33 +a 42 +a 43 = =7 322 Properties of a double sum ( ) ( ) a j a i = j= i= Forexampleleta= [ c d e ] The = a 2 i +2 i= i<j a 2 i + i= i =j a i a j a i a j (30) [c +d+e] [c +d+e] = c 2 +d 2 +e 2 +2cd +2ce +2de

7 INTRODUCTION TO MATRIX ALGEBRA 7 4 MATRIX OPERATIONS 4Scalarmultiplicatio(matrix)GiveamatrixAadascalar λ,theproductof λada,writte λa,isdefiedtobe λa λa 2 λa λa 2 λa 22 λa 2 λa = (3) λa m λa m2 λa m 42Scalarmultiplicatio(vector)Giveacolumvector aadascalar λ,theproductof λad a,writte λ a,isdefiedtobe λ a = Forthesecodcolumofamatrixwecouldwrite λ a 2 = λa λa 2 λa m λa 2 λa 22 λa 32 λa m2 43TraceofasquarematrixThetraceofamatrixisthesumofthediagoalelemetsadis deotedtracosiderthematrixcbelow (32) (33) C = (34) 5 2 ThetraceofCis[ ]=3 44Additioofvectors-Thesumofavectorawithmelemetsadavectorbhavigmelemets isavectorcwithmelemetsadwhoseelemetsaregiveby c j = a j +b j j (35) This gives c = c c 2 = c m a a 2 + a m b = b m a +b a 2 + a m +b m (36)

8 8 INTRODUCTION TO MATRIX ALGEBRA 45LiearcombiatiosofvectorsIfaadbaretwo-vectorsadsadtaretworealumbers, ta+sbissaidtobetheliearcombiatioofaadbisymbolswewrite, t a a 2 +s a m b b m = ta +sb ta 2 +s ta m +sb m (37) 45ExampleLet a = a adlet b = b Lett=2ads=4Theweobtai a 2 a 3 2 a a 2 +4 b = 2a +4b 2a 2 +4 = c c 2 a 3 b 3 2a 3 +4b 3 c 3 where c represets the liear combiatio c 2 c 3 b Numerical Example = = Writig a system of equatios as a liear combiatio of vectors Cosiderthreevectors,eachwithtwoelemetsCallthevectors a, a 2 ad bcalltheelemets ofthefirstoea ada 2,theelemetsofthesecodoea 2 ada 22 adtheelemetsof b,b ad Nowcosidertwoscalarsdeotedx adx 2 Nowmultiply a byx ad a 2 byx 2 adaddthe products We obtai x ( a a 2 ) +x 2 ( a2 a 22 ) Ifsetthisexpressioequalto bweobtai = ( ) a x a 2 x + ( ) a x +a 2 x 2 = a 2 x +a 22 x 2 ( ) a2 x 2 a 22 x 2 = ( ) a x +a 2 x 2 a 2 x +a 22 x 2 whichisaliearsystemof2equatiosi2ukows Wecawriteageeralsystemofm equatios i ukows as ( b ) (38) (39) x a +x 2 a 2 + +x a = b (40) wherex i areaseriesofscalarukowsadeacha j isacolumoftheamatrixofcoefficiets

9 INTRODUCTION TO MATRIX ALGEBRA 9 46AdditioofmatricesThesumCofamatrixAhavigmrowsadcolumsadamatrixB havigmrowsadcolumsisamatrixhavigmrowsadcolumswhoseelemetsaregive by This gives c ij = a ij +b ij i,j (4) c c 2 c a a 2 a b b c 2 c 22 c 2 C = = a 2 a 22 a c m c m2 c m a m a m2 a m b m b m2 b m a +b a 2 + a +b a 2 + a a 2 + = a m +b m a m2 +b m2 a m +b m For example = Ier(dot)productoftwovectorsTheier(scalarordot)producttotwovectorsu,vof legthisthescalarquatitydeotedby u v = u i v i = u v +u 2 v 2 + +u v (43) i= Itiseasiesttoseehowtomultiplytwovectorsifwewritethefirstoeasarowvectoradthe secodoeasacolumvectorforexample [ ] = = = 48 (44) 5 48MultiplicatioofamatrixadacolumvectorWecamultiplyamatrixadcolumvector ifthematrixhasthesameumberofcolumsasthereareelemetsithecolumvectortheresult ofthismultiplicatioisacolumvectorwiththesameumberofelemetsasthematrixhasrows Thei th elemetoftheresultigcolumvectorisobtaiedasthedotproductofthei th rowofthe matrixadthecolumvectorspecificallyforam matrixaada columvectorb, (42) For example c i = a ik b k,i =,,m (45) k= =

10 0 INTRODUCTION TO MATRIX ALGEBRA Specifically =8adsoo Or cosider the example = MultiplicatioofarowvectoradamatrixWecamultiplyarowvectoradamatrixif thematrixhasthesameumberofrowsasthereareelemetsitherowvectortheresultofthis multiplicatioisarowvectorwiththesameumberofelemetsasthematrixhascolumsthe i th elemetoftheresultigrowvectorisobtaiedasthedotproductoftherowvectoradthei th columofaspecificallyforam matrixaadam columvectorb, For example c i = m b k a ki,i =,, (46) k= ( ) = ( ) 2 3 Specifically =-2adsoo Here is a secod example ( ) = ( 2 2 ) MultiplicatioofmatricesGiveam matrixaada rmatrixb,theproductabis defiedtobeam rmatrixc,whoseelemetsarecomputedfromtheelemetsofa,baccordig to c ij = a ik b kj,i =,,m,j =,,r (47) k= Iotherwordstoobtaitheij th elemetofcwetakethei th rowofaadj th columofbad form the ier product 40 Example Cosider multiplyig the followig two matrices A ad B A = , B = 2 4 (48) WeobtaithefirstelemetoftheproductbymultiplyigthefirstrowofAbythefirstcolum ofb [ ] = c = 2 (49) 2

11 INTRODUCTION TO MATRIX ALGEBRA WeobtaithesecodelemetofthefirstrowoftheproductbymultiplyigthefirstrowofAby thesecodcolumofb [ ] 2 4 = c 2 = 5 (50) WeobtaithethirdelemetofthefirstrowoftheproductbymultiplyigthefirstrowofAby thethirdcolumofb [ ] = c 3 = (5) 3 CombiigoperatiosforthefirstrowofAadthematrixBweobtai ( ) = NowcosiderthesecodrowofAadthematrixBPerformigtheseoperatiosweobtai ( ) = Completig the operatios we obtai = Example 2 As a secod example cosider the matrices below A = B = 0 2 (52) Theelemetc comesfrommultiplyigthefirstrowofawiththefirstcolumofbasfollows: c = ( ) 2 = = 8 (53) Similarlytheelemetc 32 comesfrommultiplyigthethirdrowofawiththesecodcolumof B as follows: c 32 = ( 0 4 ) 0 = = 6 (54) 4 Multiplyig out the rest of the etries gives C = (55) 5 6 5

12 2 INTRODUCTION TO MATRIX ALGEBRA 4WritigasystemofequatiosasamatrixproductCosideram matrix,a vector adam vector Thecaseofasquarematrixishadledbysettigm= CallthematrixA adthevectors xad bcalltheelemetsof x,x,x 2,,x,adtheelemetsof b,b,,,b CosideracasewhereAisa4 3matrixadxisa3 vectoradbisa4 vectormultiplythe matrixabythecolumvectorxadsetitequaltothevectorbasfollows a a 2 a 3 a 2 a 22 a 23 x a 3 a 32 a 33 x 2 = x a 4 a 42 a 3 43 If we the carry out the multiplicatio we obtai b b 3 b 4 (56) a x +a 2 x 2 +a 3 x 3 =b a 2 x +a 22 x 2 +a 23 x 3 = a 3 x +a 32 x 2 +a 33 x 3 =b 3 (57) a 4 x +a 42 x 2 +a 43 x 3 =b 4 whichisaliearsystemof4equatiosi3ukowsthegeeralsystemofmequatiosi ukows ca be writte a x + a 2 x a x = b a 2 x + a 22 x a 2 x = a 3 x + a 32 x a 3 x = b = (58) a m x + a m2 x a m x = b m Ithissystem,thea ij sadb i saregiverealumbers;a ij isthecoefficietfortheukowx j ithei th equatiowecallthesetofalla ij sarragediarectagulararraythecoefficietmatrix ofthesystemusigmatrixotatiowecawritethesystemas a a 2 a 2 a 22 a 3 a 32 a m a m2 Cosider the followig matrix A ad vector b Wecathewrite A = a a 2 a 3 a m Ax =b x x 2 = x , b = b b m (59) 3 8 (60) 4

13 INTRODUCTION TO MATRIX ALGEBRA 3 for the liear equatio system Ax =b x x 2 = x 3 (6) x +2x 2 +x 3 = 3 2x +5x 2 +2x 3 = 8 3x 4x 2 2x 3 = 4 (62) 42 A system of equatios with a idetity coefficiet matrix Cosider a system of variables ad equatios The coefficiet matrix is square If the coefficiet matrix is a idetity matrix the the solutio is obvious upo ispectio Cosider the followig 3 3 example x x 2 x 3 x Ix =b = b b 3 b m (63) x x 2 = 2 2 (64) 0 0 x 3 Itisobviousthatx =-2adsooas x +0 x 2 +0 x 3 =-2IfoeweretouseGaussia elimiatiotosolveasystemofequatiosiukowsadrewritethesystemasamatrix equatioateachstep,itisclearoewouldedupwithasystemwherethecoefficietmatrixwas a idetity matrix 43 A system of equatios with a diagoal coefficiet matrix Cosider a system of variables ad equatios The coefficiet matrix is square If the coefficiet matrix is a diagoal matrix the the solutio ca be obtaied by solvig each equatio idividually by oe simple divisio a a a a x x 2 x 3 x Ix =b = Itisclearthata x =b whichimpliesthatx = b a Similarlyx 2 = a 22 adsoo Cosider the followig 3 3 example b b 3 b m (65)

14 4 INTRODUCTION TO MATRIX ALGEBRA x x 2 = 6 8 (66) Itisobviousthat(-3)x =6whichimpliesthatx = 6 3 =-2adsoo 44SomepropertiesofmatrixoperatiosLet αad βdeoterealumbers(scalars), a, b, cdeote -vectors, ad A, B, C deote matrices The properties are coditioal o the operatios beig defiedforthecaseipoit 44 Equality vectors:two-vectorsaadbaresaidtobeequalifalltheircorrespodigcompoetsare equal Equality is oly possible for vectors of the same dimesio matrices:twomxmatricesaadbaresaidtobeequalifalltheircorrespodigcompoets are equal Equality is oly possible for matrices of the same dimesio 442 Multiplicatio by a scalar a:( α+β)a=αa+ βa b: α(a+b)=αa+αb c: α(βa)=(αβ)a NotethatAadBabovecabereplacedbyaadbasi()(a)=a 443 Additio a: a + b = b + a b: a +0 = a c: ( a + b) + c = a + ( b + c) d: a + ( a) =0 e: A +B =B +A f: A + (B +C) = (A +B) +C g: A +0 =0 +A =A h: A + ( A) =0 444 Multiplicatio a: a b = b a b:ab =BA c:a(bc)=(ab)c d: α( b + c) = α b + α c e: A(B +C) =AB +AC f: (B +C)A =AB +CA g: (α a) b = a(α b) = α( a b) h: a a >0 a =0 i: a 0 =0 a =0 j: A0 =0A =0 k: AI =IA =A 445 Trasposes a: (A ) =A b: (ABC) =C B A c: (A +B) =A +B x 3

15 INTRODUCTION TO MATRIX ALGEBRA Properties of the trace a:trace(i)= b:trace(abc)=trace(cab)=trace(bca) c:trace(a+b)=trace(a)+trace(b) d:tr(ab)=tr(ba)ifbothabadbaaredefied e:tr(ka)=ktr(a)wherekisascalar 45 Idempotet matrices- A matrix is called idempotet if A 2 = A (67) For example the idetity matrix is idempotet Cosider the matrix M below M = (68) We ca verify that it is idempotet by carryig out the multiplicatio MM = Cosider the multiplicatio of the first row ad first colum 08 ( ) = = 08 (70) Or cosider the multiplicatio of the first row ad secod colum 02 ( ) = = 02 (7) LaterwewilldiscussaimportatcoceptcalledtherakofamatrixForaidempotetmatrix A,tr(A)=rakofA (69)