MATH 311-504Topics in Applied MathematicsLecture 8:Matrix algebra (continued).MatricesDefinition. An m-by-n matrix is a rectangulararray of numbers that has m rows and n columns:a11a12. . . a1na21a22. . . a2n............am1am2. . . amnNotation: A = (aij)1≤i≤n, 1≤j≤mor simply A = (aij)if the dimensions are known.Matrix additionDefinition. Let A = (aij) and B = (bij) be m×nmatrices. The sum A + B is defined to be the m×nmatrix C = (cij) such that cij= aij+ bijfor allindices i, j.That is, two matrices with the s ame dimensions canbe added by adding their corresponding entries.a11a12a21a22a31a32+b11b12b21b22b31b32=a11+ b11a12+ b12a21+ b21a22+ b22a31+ b31a32+ b32Scalar mul ti plicationDefinition. Given an m×n matrix A = (aij) and anumber r , the scalar multiple rA is defined to bethe m×n matrix D = (dij) such that dij= raijforall indices i, j.That is, to multiply a matrix by a scalar r,one multipli es each entry of the m atrix by r .ra11a12a13a21a22a23a31a32a33=ra11ra12ra13ra21ra22ra23ra31ra32ra33The m×n zero matrix (all entries are zer os) isdenoted Omnor simply O.Negative of a matrix: −A is defined as (−1)A.Matrix diff erence: A − B is defined as A + (−B).As far as the linear operations (addition and scalarmultiplication) are concerned, the m×n matricescan be regarded as mn-dimensional vectors.Matrix multiplicationThe product of matrices A and B is defined if thenumber of columns in A matches the number o frows in B.Definition. Let A = (aik) be an m×n matrix andB = (bkj) be an n×p matrix. T he product AB isdefined to be the m×p matrix C = (cij) such thatcij=Pnk=1aikbkjfor all indices i, j.That is, matrices are multiplied row by column:∗ ∗ ∗* * *∗ ∗* ∗∗ ∗ * ∗∗ ∗ * ∗=∗ ∗ ∗ ∗∗ ∗* ∗A =a11a12. . . a1na21a22. . . a2n............am1am2. . . amn=v1v2...vmB =b11b12. . . b1pb21b22. . . b2p............bn1bn2. . . bnp= (w1, w2, . . . , wp)=⇒ AB =v1·w1v1·w2. . . v1·wpv2·w1v2·w2. . . v2·wp............vm·w1vm·w2. . . vm·wpAny system of linear equations can be representedas a matrix equation:a11x1+ a12x2+ · · · + a1nxn= b1a21x1+ a22x2+ · · · + a2nxn= b2· · · · · · · · ·am1x1+ am2x2+ · · · + amnxn= bm⇐⇒ Ax = b,whereA =a11a12. . . a1na21a22. . . a2n............am1am2. . . amn, x =x1x2...xn, b =b1b2...bmProperties of matrix multiplication:(AB)C = A(BC ) (associative law)(A + B)C = AC + BC (distributive law #1)C (A + B) = CA + CB (distributive law #2)(rA)B = A(rB) = r(AB)Any of the above identities holds provided thatmatrix sums and products are well defined.If A and B are n×n m atrices, then both AB and BAare well defined n×n matrices.However, in general, AB 6= BA.Example. Let A =2 00 1, B =1 10 1.Then AB =2 20 1, BA =2 10 1.If AB does equal BA, we say that the matrices Aand B commute.Problem. Let A and B be arbitrary n×n matrices .Is it true that (A − B)(A + B) = A2− B2?(A − B)(A + B) = (A − B)A + (A − B)B= (AA − BA) + (AB − BB)= A2+ AB − BA − B2Hence (A − B)(A + B) = A2− B2if and only ifA commutes with B.Diagonal matricesIf A = (aij) is a square matrix, then the entries aiiare called diagonal entries. A square matrix iscalled diagonal if all non-diagonal entries are zeros.Example.7 0 00 1 00 0 2, denoted diag( 7, 1, 2).Let A = diag(s1, s2, . . . , sn), B = diag(t1, t2, . . . , tn).Then A + B = diag(s1+ t1, s2+ t2, . . . , sn+ tn),rA = diag(rs1, rs2, . . . , rsn).Example.7 0 00 1 00 0 2−1 0 00 5 00 0 3=−7 0 00 5 00 0 6Theorem Let A = diag(s1, s2, . . . , sn),B = diag(t1, t2, . . . , tn).Then A + B = diag(s1+ t1, s2+ t2, . . . , sn+ tn),rA = diag(rs1, rs2, . . . , rsn).AB = diag(s1t1, s2t2, . . . , sntn).In particular, diagonal matrices always commute.Example.7 0 00 1 00 0 2a11a12a13a21a22a23a31a32a33=7a117a127a13a21a22a232a312a322a33Theorem Let D = diag(d1, d2, . . . , dm) and A bean m×n matrix. Then the matrix DA is obtainedfrom A by multiplying the ith row by difori = 1, 2, . . . , m:A =v1v2...vm=⇒ DA =d1v1d2v2...dmvmExample.a11a12a13a21a22a23a31a32a337 0 00 1 00 0 2=7a11a122a137a21a222a237a31a322a33Theorem Let D = diag(d1, d2, . . . , dn) and A bean m×n matrix. Then the matrix AD is obtainedfrom A by multiplying the ith column by difori = 1, 2, . . . , n:A = (w1, w2, . . . , wn)=⇒ AD = (d1w1, d2w2, . . . , dnwn)Identity matrixDefinition. The identity matrix (or unit matrix) isa diago nal matrix with all diagonal entries equal to 1.The n×n identity matrix is denoted Inor simply I .I1= (1), I2=1 00 1, I3=1 0 00 1 00 0 1.In general, I =1 0 . . . 00 1 . . . 0............0 0 . . . 1.Theorem. Let A be an arbitrary m×n matrix.Then ImA = AIn= A.Matrix polynomialsIf B is not a square matrix then BB is not defined.Definition. Given an n-by-n matrix A, letA2= AA, A3= AAA, . . . , Ak= AA . . . A|{z }k times, . . .Also, let A1= A and A0= In.Associativ i ty of matrix multiplication implies that all powersAkare well defined and AjAk= Aj+kfor all j, k ≥ 0. Inparticular, all powers of A commute.Definition. For any poly nomialp(x) = c0xm+ c1xm−1+ · · · + cm−1x + cm,letp(A) = c0Am+ c1Am−1+ · · · + cm−1A + cmIn.Example. A =2 11 1.A2= AA =2 11 12 11 1=5 33 2,A3= A2A =5 33 22 11 1=13 88 5,A4= A2A2=5 33 25 33 2=34 2121 13.By the way, 1, 1, 2, 3 , 5, 8, 13, 21, 3 4, . . . arefamous Fibonacci numbers given by f1= f2= 1and fn= fn−1+ fn−2for n ≥ 3.Example. p(x) = x2− 3x + 1, A =2 11 1.p(A) = A2− 3A + I =2 11 12− 32 11 1+1 00 1=5 33 2−6 33 3+1 00 1=0 00 0.Thus A2− 3A + I = O.Properties of matrix polynomialsSuppose A is a square matrix, p(x), p1(x), p2(x) arepolynomials, and r is a scalar. Thenp(x) = p1(x)+p2(x) =⇒ p(A) = p1(A) + p2(A)p(x) = rp1(x) =⇒ p(A) = rp1(A)p(x) = p1(x)p2(x) =⇒ p(A) = p1(A)p2(A)p(x) = p1(p2(x)) =⇒ p(A) = p1(p2(A))In particular, matrix polynomials