Background B.6 Vectors and Matrices 35 A matrx withm rowsandscalled amrix of the order()oran n)matrix.For the special case where m=n,the matrix is calle 08q11ar B.6 Vectors and Matrices An entlty specified byn numbers in a certain order (ordered n-tuple)is an It should be stressed at this point that a matrix is not a nu r such as a enient n-dim ector.Thus,an ordered n-tuple (1,r3.rn)represents an determinant,but an array of numbers arranged in a particua n-dimensional vectox.Vectors may be represented as a row (row vector): to abbreviate the representation of matrix A in Eq.(B. 50)with t implyng oforderth ement. x1z2.zn] the nota to ()Note that the first index i of ain e row and the s cond index i or as a column (column vector): indicates the column of the elemer toueqtions (B.)may now be expressedin symboli form y Ax (B.51) x= 的 [11412 41n n Simultaneous linear equations can be viewed as the transformation of one vector (B.52 into another.Consider,for example,the n simultaneous linear equations a11z1+a12r2++a1zn Laml am2amn」LnJ y2=2111+g222十.+@2mxm represen (B.48) aBT efie such an operation m=am1z1+am2r2十.+cmnn B.6-1 Some Definitions and Properties Aquare matrix whose elements are where except on the main diag- If we define two column vectors x and y as onal is a diagonal matrix.An example of a diagonal matrix is r2001 (B49) 010 L005] A diagonal matrix with unity for all its diagonal elements is called an identity on that transforms matrix or a unit trix denoted by I.Note that this is a square matrix: then Egs.(B.48)may be viewed as the rela vector x into vector y.Such a transformation is c 100 01 dafina the a of vectors.In order to perform a of coeficienta i appearing in Eas.(B.48).This array matrix and is 010·0 denoted by A for convenence I= 0010 (B.53) fa11。a12+a1a1 21422a2n (B,50) L000,1J A■ The order of the unit matrix is sometimes indicated Thus.I represents the nxn unit matrix (or identi hall omit the
37 36 Background B.6 Vectors and Matrices A matrix having all its elementsr isaero matrix. 「(a11+)(a1z+b12). (ain+bim) matrix A is a symmetric matrix(symmetry about the (a2m+b21} a2+ba)t(a2n+b2n】 of the same order are said to be equal if they are equal element A+B= by element.Thus, A=()mxn and B=()mxn L(am1+ml）(am2+bn2(amh+bnn) then A =B only if a=ba for all i and j. If the oad com of an nged so that the A+B=(aij+big)mxn elements n the ithrow now become the elements of the ith 6A文c Aand is denot Note that two matrices can be added only if they are of the same order. 2.Multiplication of a Matrix by a Scalar 「211 We define the multiplication of a matrix A by a scalar eas A=32 eA-[23 [ca11 ca1z cain 123 ca21 ca22.c2m L13」 Thus,if A=(amxn then AT=(ajt)nxm (B.54) 3.Matrix Multiplication We define the product Note that AB=C (AT)T=A (B.55) in which the eleme ofC in and found by adding the products of the eleme ts of A in the ith row with the corresponding elements B.6-2 Matrix Algebra of B in the jth column.Thus, cg=aub1y+aub2对+.+anbn Weshalnowdeinematrtoperationa,such吃aditionsubtactiom,mltp (B.56 cation,and division of matrices.The ions should be formulated that they are useful in the manipulation of matrices. This result is shown below. 1.Addition of Matrices For two matrices A and B.both of the same order (mxn) b1112 .虹 421a2a2z 21b22. A= and B= A(mxn) C(mxp) B(nxp ve define the sumn A+B as
Background B.6 Vectors and Matrioes 30 Note carefully that the number of columns of A must be equal to the number of AI=IAA (B.60) oduet of matris rows of B it this procedure A and B.is oduct AB is not defined and is 2ne地oe1aem时 nding product. mal to the number of rows of mable to matrix B for the product AB.Observe 4.Multiplication of a Matrix by a Vector and B matrix,A and B are conformable asents Eq.(B.48).The right-hand nd cis a Coide themr4:（B52,whi由rept花ot side of Eq.(B.52)is a prod time being,we tre nx I matrix,then the product rule,yields the right-hand side of Eq 「231 89571 Ax,according to the mat (B.48).Thus,we may trix by a vector by treating the vector as if it 11 13121 =3423 2111 were an nx 1 r aint of conformability still applies.Thus, d is L31 51047 in this case,xA is not defined 5.Matrix Inversion T21 To define the inverse of a matrix,let us consider the set of equations [2131=8 「a11a12+41r 21a22 . (B.61 We can solve this set of equations for ,.in terms of yi,a.n by 3 using Cran er's rule see Eq.(B.31)].This yields the matricesare no longer conformable for the product.It is evident that in general, 「脚脚 1 「1 AB≠BA B.62) Indeed,AB may exist and BA may not exist,or vice versa,as in the above exam. ples.We shall see later that for some special matrices. 赞. AB=BA (B57 When Eq.(B.57)is true,matrices A and B are said to commute.We must stress in the mat r of element aa is given by (-1)t times the mtrice do not come Opertion(n determin e( .1)matrix that is obtained when the ith row and are deleted. atrix product AB,matrix A is said to be postmultiplied by B or premultiplied by A.We may also verify the following We can express Eq.(B.61)in matrix form as said to be y=Ax (B.63 (A+B)C=AC+BC (B.58) We can now define A-1,the inverse of a square matrix A,with the property (B.64) C(A+B)=CA+CB B.59) A-1A=I (unit matrix) Then,premultiplying both sides of Eq.(B.63)by A-1,we obtain We can verify that anymatrix A premultiplied or postmuitiplied by the identity matrix I remains unchanged: A-ly=A-Ax=Ix=x
40 Background B.6 Vectors and Matrices or B.6-3 Derivatives and Integrals of a Matrix x=A-ly (B.65) Elements of a matrix need not be constants;they may be functions of a variable. A comparison of Eg.(B.65)with Eg.(B.62)shows that For example,if Du Dal·Dl门 「e-4sint】 A-e+ (B.68) A1- (B.66) then the matrix elements are functions of t.Here,it is helpful to denote A by A(t). Also,it would be helpful to define the derivative and integral of A(t). The derivative of a matrix A(t)(with respect to is defined as a matrix whose fith element is the derivative (with respect to t)of the ijth element of the matrix One of the conditions .(B.61)is that the tal the that the A.Thus,if solutio the A(t)=lag(mxn ondition that the determinam of the matrix be onzero.A matrix whose determinant is nonzero is a nonsin- then huaveriston foroningular (qure)mtrix.By Ao=[层t利 (B.69a) A-A=I (B.67a) mioigt恤ogaaayA-'adthapmeanipigt好A.ecoahow A(日=[a(t)mxn (B.69%) Thus,the derivative of the matrix in Eq.(B.68)is given by AA-1=I (B.67b 「-2e-2x Note that the matrices A and A-1 commute. A(e)= -e-4-2e-2t t of the matrix A: A=123 「A0=(∫） (B.70) L321 Here Thus,for the matrix A in Eq.(B.68),we have [Dul=-4.[Dl=8.[Di3l [∫e-rdt∫sin dt1 Dl=1,D2l=-1,D2l=-1 A()t= ∫e'd4je-÷+2e-24)d Dstl=1.ID32!=-5,D313 We can readily prove the following identities and Al-4.Therefore, [-411 A+副-会+ (B71a) -4-13 c=绘 (B.71b) er ofto the uber of 票AB)-验B+A里-AB+AB (B.71e)
Background B.6 Vectors and Matrices 48 The proofs of identie(B.71a)and (B.71b)are trivial We can prove E (B.71c) ows.Let A be an mxn matrix and B an nxp matrix.Then,if a202-A C=AB =0 (B.76b) from Eq.(B.56),we have anl Equation (B.76a)or (B.76b)Is known as the characteristie equation of the matrix A and can be expressed as and ia-+ (B.72) Q()=A1-A1=X"+4n-1An-1+.+a1A+eoA0=0 B771 4 (is the characteristic polynomial of.Theof the characte nvalues of A and,corresponding to each ■d+k eigenvalue,ther ctor that satisfies Eq.(B.74). The Cayley. the em states that every n x n matrix A satisfies Equation (B.72)along with the multiplication rule clearly indicate that di is the ikth element of matrix AB and eik is the ikth element of matrix AB.Equation Dy A: B.71c)then follows If we let B =A-1 in Eq.(B.7Ic),we obtain Q{A)=A+an-1An-1+.+a1A+a0A0=0 (B.78) 盖AA-验A+AA Functions of a Matrix But since The Cayley-Hamilton theorem can be used to evaluate funetions of a square aA-==0 matrix A,as shown below. Consider a function f()in the form of an infinite power series we have B.79) A-A验A (B.23 )=a+aA+2号+.+= BecauseA satisfies the characteristic Eq(B.77),we can write .hi Ma The am an=-an-idn-1 -cn-a4n-2-.-013-00 (B.80 For an (n)square matrix A,any vector x(x)that satisfies the equation If we muti iply both sides bythe left-hand side isand the right-hand side A.Using Eq.(B.80),if Ax=Ax (B.74) ale cr hrte ah( ing in this】 (A-x=0 (B.75) ()=两+A+A2+.+-1An- (B81 The slution for thisset of homogeneous equations exists if and only if ne that there are n distinet eigenvalues入，Az then Eq.(B.81) A-I=-Aj0 (B.76a n simultaneous quations