Matrices and Linear Algebra - Texas A&M University

1 Chapter 2 Matrices and Linear BasicsDefinition anm narray of scalars from a givenfieldF. The individual values in the matrix are ^213 124 B=^1234 Thesizeof the array is written asm n,wherem ncAnumber of rows number of columnsNotationA= amn A rowstAAccolumnsA:= uppercase denotes a matrixa:= lower case denotes an entry of a matrixa matrices3334 CHAPTER 2. Matrices AND Linear Algebra (1) Ifm=n, the matrix is (1a) A matrixAis said to bediagonalifaij=0iW=j.(1b) A diagonal matrixAmay be denoted by diag(d1,d2,.. ,dn)whereaii=diaij=0jW= diagonal matrix diag(1,1,.. ,1) is called theidentitymatrixand is usually denoted byIn= or simplyI,whennis assumed to be known.

2 0 = diag(0,.. ,0)is called thezero matrix.(1c) A square matrixLis said to belower triangulariffij=0i<j.(1d) A square matrixUis said to be upper triangular ifuij=0i>j.(1e) A square matrixAis calledsymmetricifaij=aji.(1f) A square matrixAis calledHermitianifaij= aji( z:= complex conjugate ofz).(1g)Eijhas a 1 in the (i, j) position and zeros in all other positions.(2) A rectangular matrixAis callednonnegativeifaij 0alli, is calledpositiveifaij>0alli, of these Matrices has some special properties, which we will studyduring this BASICS35 Definition set of allm nmatrices is denoted byMm,n(F),whereFis the underlyingfield (usuallyRorC).

3 In the case wherem=nwe writeMn(F) to denote the Matrices of sizen ,nis a vector space with basis given byEij,1 i m,1 j , Addition, MultiplicationDefinition matricesAandBare equal if and only if they havethesamesizeandaij=bijalli, any matrix and Fthen the scalar multipli-cationB= Ais defined bybij= aijalli, Matrices of the same size then thesumAandBis defined byC=A+B,wherecij=aij+bijalli, jWe can also compute thedifferenceD=A Bby summingAand ( 1)BD=A B=A+( 1) addition inherits many properties from , B, C Mm,n(F)and , F,then(1)A+B=B+Acommutivity(2)A+(B+C)=(A +B)+Cassociativity(3) (A+B)= A+ Bdistributivity of a scalar(4) IfB=0(a matrix of all zeros) thenA+B=A+0=A(4)( + )A= A+ A36 CHAPTER 2.

4 Matrices AND Linear Algebra (5) ( A)= A(6)0A=0(7) 0= Rn,x=( )y=( ).Then the scalaror dotproduct ofxandyis given by x, yX=n3i= (i) Alternate notation for the scalar product: x, yX=x y.(ii) The dot product is defined only for vectors of the same (1,0,3, 1) andy=(0,2, 1,2) then x, yX=1(0) + 0(2) + 3( 1) 1(2) = nandBisn (A) denote the vectorwith entries given by theithrow ofA,andletcj(B) denote the vector withentries given by thejthrow ofB. The productC=ABis them pmatrixdefined bycij= ri(A),cj(B)Xwhereri(A) is the vector inRnconsisting of theithrow ofAand similarlycj(B) is the vector formed from thejthcolumn ofB.

5 Other notation forC=ABcij=n k=1aikbkj1 i m1 j }101321]andB= 2130 11 .ThenAB=}1211 4]. BASICS37 Properties of matrix multiplication(1) IfABexists, does it happen thatBAexists andAB=BA?Theanswer is usually no. FirstABandBAexist if and only ifA Mm,n(F)andB Mn,m(F). Even if this is so the sizes ofABandBAare different (ABism mandBAisn n) unlessm= even ifm=nwe may haveABW= They may be different sizes and if they are the same size ( )theentriesmaybedifferentA=[1,2]B=} 11]AB=[1]BA=} 1 212]A=}1234]B=} 1101]AB=} 13 37]BA=}2234](2) IfAis square we defineA1=A, A2=AA, A3=A2A=AAAAn=An 1A=A A(nfactors).

6 (3)I= diag(1,.. ,1). IfA Mm,n(F)thenAIn=AandImA= (Matrix Multiplication Rules).AssumeA, B,andCare Matrices for which all products below make sense. Then(1)A(BC)=(AB)C(2)A(B C)=AB ACand(A B)C=AC BC(3)AI=AandIA=A(4)c(AB)=(cA)B(5)A0=0and 0B=038 CHAPTER 2. Matrices AND Linear Algebra (6) ForAsquareArAs=AsArfor all integersr, s :IfACandBCare equal, it does not follow thatA=B. See use an alternate notation for matrix entries. For anymatrixBdenote the (i, j)-entry by (B) Mm,n(F).(i) Define thetransposeofA, denoted byAT,tobethen mmatrixwith entries(AT)ij=aji.(ii) Define theadjointofA, denoted byA ,tobethen mmatrix withentries(A )ij= ajicomplex conjugateExample }123541]AT= 152431 In The rows ofAbecome the columns ofAT, taken in the sameorder.

7 The following results are easy to (Laws of transposes).(1)(AT)T=Aand(A ) =A(2)(A B)T=AT BT(and for )(3)(cA)T=cAT(cA) = cA (4)(AB)T=BTAT(5) IfAis symmetricA= BASICS39(6) IfAis HermitianA=A .More facts about (1) We know (AT)ij= ((AT)T)ij= (AT)T=A.(2) (A B)T=aji (A B)T=AT (1)Ais symmetric if and only ifATis symmetric.(1) Ais Hermitian if and only ifA is Hermitian.(2) IfAis symmetric, thenA2is also symmetric.(3) IfAis symmetric, thenAnis also symmetric for matrix is called skew-symmetric ifAT= matrixA= 012 10 3 23 0 is (1) IfAis skew symmetric, thenAis a square matrixandaii=0,i=1.

8 ,n.(2) For any matrixA Mn(F)A ATis skew-symmetric whileA+ATis symmetric.(3) Every matrixA Mn(F)can be uniquelywritten as the sum of askew-symmetric and symmetric (1) IfA Mm,n(F), thenAT Mn,m(F). So, ifAT= Awemust havem= aiifori=1,.. , 2. Matrices AND Linear Algebra (2) Since (A AT)T=AT A= (A AT), it follows thatA ATisskew-symmetric.(3) LetA=B+Cbe a second such decomposition. Subtraction gives12(A+AT) B=C 12(A AT).The left matrix is symmetric while the right matrix is both are the zero (A+AT)+12(A AT). 110ois skew-symmetric. LetB=}12 14]BT=}1 124]B BT=}03 30]B+BT=}2118].ThenB=12(B BT)+12(B+BT).

9 An important observation about matrix multiplication is related to ideasfrom vector spaces. Indeed, two very important vector spaces are associatedwith Mm,n(C).(i)Denote bycj(A):=jthcolumn ofAcj(A) Cm. We call the subspace ofCmspanned by the columns ofAthecolumn (A),..,cn(A) denoting the columns BASICS41the column space isS(c1(A),..,cn(A)).(ii) Similarly, we call the subspace ofCnspanned by the rows (A),..,rm(A) denoting the rows ofAthe row spaceis thereforeS(r1(A),..,rm(A)).Letx Cn,whichweviewasthen 1matrixx=[ ] defined andAx=n3j=1xjcj(A).That is to say,Ax S(c1(A),.. ,cn(A)) =columnspace Mn(F).

10 The matrixAis said to beinvertibleif there is a matrixB Mn(F) such thatAB=BA= this caseBis called theinverseofA, and the notation for the inverse isA (i) LetA=}13 12]ThenA 1=15}2 311].(ii) Forn=3wehaveA= 12 1 13 1 23 1 A 1= 01 1 13 2 37 5 A square matrix need not have an inverse, as will be discussed in thenext section. As examples, the two Matrices below do not have inversesA=}1 2 12]B= 101021122 42 CHAPTER 2. Matrices AND Linear Linear SystemsThe solutions of Linear systems is likely the single largest application of ma-trix theory. Indeed, most reasonable problems of the sciences and economicsthat have the need to solve problems of several variable almost without ex-ception are reduced to component parts where one of them is the solutionof a Linear system.