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Operations of, h cIctnent, , i, , one, , t", , 1.2, s, , . 2. Addition and, , (ion, 'j lic two, they have, two, A, , of, , Ij are, A, (f,e. •..•nne, and p u, , be, , o', , . of order, , h'. h, , the, , A, , orrey»oncling elctncnt,sO/A and IJ, , o/, , B • C if and only, , of order, each element, , )for all values of i and j,, , of two matrices A and l/ is defined as the matrix t) 2 (d I, , Sitiillarly. lhe, , lement of, IS the, —, A B z t) if and only, , of the correspondtng elements of A and, all, , =d, , i.e., , ot different orders can not be added or, addition obeys following properties, , Note., , two, , cotntnutative law holds for matrtcrs i.e.,, , (O, , A, , or, , z, , 4, , 4, , =, , i.e.,, (y) The assoctative law holds for matrtx (1/1/1ition, The matrtccs obey distributive law of addition i.e.., , where, , js any nutnbcr., , and is any complex number, IS a matrtx,whose cach, , Slultiplication of a matrix by a scalar. If A la, , (sca ar). then LA dcfincd, corresponding cfcment of A., , then, , For example. If, "Rail, , Xa12 Raj A, 22, , matrices A = la„l and B, two, The, Matrices., of, 4. Multiplication, the number of_golumnsonA is, , multiphcd, , the order, , the matrices, of rows jn Il..tJnderfifus condition., h) and matrix is of order, If, are given by, order tn x n. Its elements, , (bol maybe, to, mu)oplivaopw, , n then, , as a scalar product of the, formed, is, AB, =, C, of columnsh as, number, c, i.e. i) element of, same, the, has, A, the, that, l. 2.., withfiiié-j, index k takes on all the values, dummy, •nie, has rows)., bij + a,2b2, +, Cig=

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125, , Matrices, , Obviously, the dummy index k tnay be replaced by any other symbol that is already not in, eqn. (I). Eqn. (1) represents the method of matrix multiplication,, usewithoutaltering two matrices :, As an, Illustration,consider, , bli, , atl an, A = an, , an, , and B =, 21, , 1>22, , and the matrix B is 2 x 2 matriF, therefore the product of, is'ü)matrix, these, ThematrixA, 2 matrix, given by, x, 3, a, IS, matrices, two, a 12, (121, a 31, , bil, , b 12, , 1221, , 1222, , (122, , C21, , alibi 1 +a12b21, C22 =, +a22b21, , a21b12+t122b22, , C31, , C32, , (1311)12 + C132b 22, , C12, , ali b ii + (1121221, , where, , C12= al 11212, + 0112/022, C21 = am b li + (1221222, C22 = (121/)12 + (122/223, C31, , C32=, , + a.32b22, , Theseelements of c are obtained from eqn. (l)., , hroperties of matrix multiplication, ("Premultiplication and Postmultiplication. In the product AB of two matrices A and B,, it is said that the matrix ISpremu tip te bYůand the matrix A is, the productAB, the matrix A is called tlypygfactor and B the postfactor., , In general, the matrix multiplication is not commutative i.e., , For example, if, Then, , A=, and, , Thu AB BA. In fact, for a given pair of matrices A and B, the products AB and BA may, not be even comparable. For example, if A be a 2 x 3 matrix and B be a 3 x 2 matrix, then AB, would be a 2 x 2 matrix, while BA would be a 3 x 3 matrix., , It may also happen that for a given pair of matrices A and B, the product AB may be, , may not be conformable. For example, if A be a 2 x, conformable(defined), but the pro uct, 3 matrix and B be a 3 x 4 matrix, then AB would be a 2 x 4 matrix, but BA can not be defined., The statement that in general the matrix multiplication is not commutative, does not mean, that we can never have AB = BA; but it simply means that AB = FA is not alwa s true. If in, anyspecial case AB = BA, then the matrices A and B are sai to commute.

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Mathematical, , 126, , Physica, , Associative law o multiplication: The multiplication of matrices is associative i.e if, for the products A(BC) and (AB)C to be defined, then, , the tuatrtces, , Proof. Let, q tvspectively., , e o suitable order, A(BC) (AB)C, i} and C = [co] be three matrices of orders m x n, n XP andp k, (aul,, , If A is an m x n matrix and b is an n Xp matrix, therefore the product AB is m XP, , matrix. The (i, k)th element of, AB =, , am. bhk {refer eqn. (1)), , As AB is m x p matrix and C is p x q matrix, therefore the product (AB) C is m Xq, matrix whose (ij) thelement is, ail: bilk Ckj =, , Again, as B is an n x p matrix and C is ap x q matrix, therefore the product BC is an (nx, element is, q) matrix whose, b hk Ckj, , As A is m x n matrix and BC is n x q matrix, therefore the product A (BC) is m Xq, matrix whose (i, j) th element is, b hk Ckj, , Comparing (2) and (3) we see that (i,j) th elements of (AB) C and A(BC) are equal for all, values of i and j (i = 1, 2, ... m; j = 1, 2,...q). Thus we have, A(BC), 'The multiplication of matrices is distributive with, 1 ti, Distributive law, regard to addition i.e., where A, B and C are the matrices of suitable orders pf the above equation to be meaningful., Proof. Let A = [a ijI mXn', ijnxp and C = [cij]n XP, As B and C are both n x p matrices, therefore (B + C) is an n x p matrix and consequently, , A (B + C) is an (m x p) matrix., Again, since A is an m x n matrix and B is a n x p matrix, therefore AB is an m XP, matrix. Similarly reasoning shows that AC is an m x p matrix., Thus, we see that L.H.S. and R.H.S. of eqn. (4) represent the matrices of the sameorder, (m x p). Now we shall show that their corresponding elements are equal., The (i, j) th elements of A (B + C), aih'Chj, , The (ij)th element of, , The (i, j) thelement of

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127, , )tatn.vs, Chi, , eletnentof, , (i., , (6), , comparingequations (5) and (6) we sce that (i, j)'h elements of A (B + C) and AB + AC, , all values of i and j ( i = 1, arcequalfor, , 2, , Ex. 1. Given A -3, , p). *Ihuswe have, , 123, , -1 andB=, , 2, , 4, , 6, , ComputeAB and BA and hence show that AB BA, Solution.As A is a 3 x 3 matrix and B is 3 x 3 matrix, therefore the products AB and BA, , are 3 x 3 matrices., , 1, , -1, , 1, , 2, 1, , -2, , -l, o, , 1, , 2, , 3, , 2, 1, , 4, 2, , 6, 3, , (-2)x2+1x4+0x2, 0., o, 0, , (-2)x3+1x6+0x3, , 0, , o, o, o, 1, , 2, , 3, , 1, , 2, 1, , 4, 2, , 6, 3, , -3, -2, , -1, , 1, , 2, -1, , -1, , 1)+2x2+3x1, 1)+2x2+3x1, -11, , = -22, , -Il, , 12 -2 . Hence AB * BA., , 2•3.Sub-matrices, Sub-matrix. The matrix of elements which remain after deleting any number of rows and, columnsof a matrixA is called the sub-matrix of A. For example the matrix, matrixof, , is a sub-

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Mathematical, , 128, , Physiq-, , and third row., because it can be obtained from the latter by deleting first column, , Princi 'al sub-matrix. If the diagonal,elementsof a s uare s b-matrixof the square, , sub-matrix is called the principal, tnatrtx A are also the la onal elements of A, t en the square, deleting., corresponding rows and, sub-matrix. rmcipal sub-matrices are obtained only, by, , columns. For example 10, , a principal sub-matrix of, 11 is, , 12, , 10, , 15, , 14, , 13, , rows and columns., because it can be obtained from the latter by deleting,first and fourth, , Leading sub-matrix. If a principal sub-matrixis obtained by deleting only someof the, , it is called the leadingsub., last%F7iiidTéQ6FFéÄF8nding columns of a square matnx A, then, , n;atrix. For example, the matrix 5, , 7 is a leading sub-matrix of, , 10, , 11, , 12, 14, , 13, , IS, , because it is obtained by deleting last (fourth) row and last (fourth) column., , 2. . Partitioning of Matrices, In some cases it is convenient to partition matrices into rectangular blocks. These blocks, containing the elements of the given matrix are defined as the sub-matrices of the given matrix., A, The dotted lines indicates the partitioning of the given matrix. Thus the partitioned matrix, al, , b2 c2, , can be written as, 12, , 21, , where, , 22

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be, matrixcan partitioned in a number of ways,, e.g.,, , 129, , al bl, , Yl, , x2 Y2, , 112, , V2, , at bl, etc., , (Il rI Ill, , PI, , Formultiplication the matrices are partitioned as follows :, , LetA = [aij I and B = [bijl be the matrices of order m XP andp x n respectively., The, in any, , arbitrary manner, then the matrix B is, A is partitioned, matrix, partitioned in such a way, thatthemultiplication of sub-matrices of both A and B exist., , Forexamplethe matrix A may be partitionedinto m submatricesof I x p and B into n, of order p x 1. As another example the matrices A and B may be partitioned into, submatrices, of indicated orders by drawing in theVdottedlines as, submatrices, , nil, , ml X pi, , nil ><P3, , PI, , All, B 21, , or, , A 22, , B 22, , 23, B31, , of A and rows of B be partitioned, In any such partitioning it is necessary that the columns, '11, may be any non-negative integers such that, in exactlythe same way; however nil, 1112,, B 21 + Al 3 B 31, , All B Il, A21 B I I, , 21, , A 22, , c 12 = c, 2, , Ex. 2. IfA=, , A 23 B 31, , All B 12 + Al 2 B 22 + Al 3 B 32, A22 B12 + A22 B22 + A23 B32, , 3, , 2, , and B = 2, , I, , 0 , then compute AB.

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130, , Mathematical, , Solution. Partitioning A and B so that, and B =, , 12, , A21 A 22, , 2, , 3, , 1, 12, , 22, , 11, , 21, , 21, , 21, , 13221, , (2, , 1, , 1, , 01, , A 21 B 12, , 2, , 12 3 11, , 11 1, , 21 1, , [1, , (4 3 31 Vo o o, , 1322, , 0 [21, , 3, , 3 11, , +, , 01, , [21, , 4, , 1 11 + 12 3, , 3, , [21, , 3, , A 22, , 4, , 2, , 4, , 21, , 2, , Special Types of Matrices, &Square Matrix. A matrix having the same number of rows as the number of columns, is calleTåÄüüäiüiiåiFii¯. For example n x n matrix, , all, (121, anl, , 22........a2n, an2, , nn, , is the square matrix,of order n or n-square matrix. In the square matrix the elementsall,, an...ann are called its diagonal elements.The sum of the diagonal elements of a squarematrix, is called the trace of, 52. Diagonal Matrix. If all the elements of a square matrix are zero except thosein the, leading, , iagonal, then the-matrix is said to be a diagonal matrix. _For example, , is a diagonal matrix of order n (i.e., n x n diagonal matrix).

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Mat'å«s, , fius an it-square matrix A, , is diagonal, , 0 for, , 131, , matrix if and only if, , i, , for i = j, , diagonal matrix of order n with diagonal, elements a, l, an....ann, is also denoted, , by, A = diag. (a, l,a22, ann), atrix. A dia onal matrix in which, 1.3., all the dia onal elements, matrix., or, are equal, say X,, scalar, example, a, c, is ca, , ...o, , .......o, is an n x n scalar matrix., , o, , o, , Thus an n-square matrix A = [a ijI, , a, Identi, , is a scalar matrix if and only if, , 0 for, , or unit matrix.iA scalar matrix in which each diagonal element is unity, is, , calledan i entity or unit matrtx., , us an n-square matrix [aijl is a, 1 for i=j, , O for, The identity matrix of order n is usually denoted by_In., , For a unit matrix I, IA = A and BI = B for arbitrary matrices A and B., , column matrix. A matrix containingonly one row or column is, —S. Row mat, a row, calleda vector. matrix avtng oneXOSS', matrix or a row vector and IS expressed as, , [all an, A matrix having one column only i.e, a matrix of orders m x I is called a column matrix, or a column vector. It is expressed as, , all, a.31, , or in order to save space all, , au ... anti, , aril, , 6. Null matrix or zero matrix, , matrix is called the null matrix or zero, the, zero,, is, matrix, n, x, m, an, of, clement, every, If, or simply by O. For example, matrix of the type m X n. It is usually denoted by O,nxn

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132, , Mathematical, , o, , 0, , 0, , 0, , o, 0, , 0 is a 3 x 4 null matrix., , o, , o, 0, , Physi, , 0, , For a null matrix O; OA = O and AO = O for an arbitrary matrix A., 7. Upper-triangulöömatnx. X'qiiåi&ifiötrix, whose every element, is callc an upper triangular matrix. Thus, , for i > j is zero, , (113......aln, , all, o, o, , (122, o, , a33......a3n, , 0, , o, , 0.......ann, , is upper-triangular matrix., , _fLower-triangular, matrix. A square matrix whose every element ao for i <j is zero,, is called the lower-triangular matrix. For example, all, , 0, , ..0, , an, , an, , o..........0, , an, , 0, , an 1, , is lower triangular matrix., , an2, , .-9. Periodic, idempotent and nilopotent matrices. A square matrix A for whichAk+l, = A, where k is a positive integer, is called the periodic matrix. If k is the least positive integer, for which Ak+l = A, then k is said to the period of matrix A., A square matrix A for whichrA2= A, is called idempotent matrix. It is obvious thatthe, idempotent matrix is the, , is an idempotentmatrix., , A square matrixA for whichAP= 0 wherep is positive integer is called the nilopotent, matrix. If p is the least positive integer for which A/'= 0, then p is said to be the indexof, matrix A., For example A =, , is a nilopotent matrix of index 2, because here A2 = 0,, , 1, , 1, , -2, , -1, , and, , 2, , 3, , 6 is a nilopotent matrix of order 3., , -3, , Ex. 3. If A and B are idempotent matrices, then A + B will be idempotent if and onlyif, , (Banaras 1998), , Solution. As A and B are idempotent matrices, we must have, A2 =A and 132 = B, , If A + B is idempotent, then, , We have

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be, , by, the, .01A and is denoted by, , 't, ot, called the, , A, , From exantple. it A, , 2, , only, , of an, and, of, symbol A' or, , then, , •• (a, It is obvious that (i. '0th element of A is the vi)th element of Ar i e. if A, Some properties of transp€xse_of.mntrix, suet AVand, tespcctivelv, then. ,, A, , be the tran vroses of matrtces, , X being any scalar (real or complex)., -€1') (LOT =, = A r +B r. A and B being conformablefor addition., (A +, A and B being conformable for multiplication., (AB)T, Proof. Let A be an m x n matrix i.e. A, •therefore, , Obviously, , Let, , As is a scalar, LA is also m n matrix given by, According to definition, (RA)T, , Again., so that, , Now obviously, , then

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Mathematical, , 134, , (c) Let A, , [aijlmxnand B, , Ph, , [bijlmxn, , = (ajilnxntand BT=, , Then, , A+B =, , Also, , Therefore (A +, , (d) Lct, , A =, , + bji]mxn, [aji + bjilnxn, [ajilnxm+ (bjilnxm, , arKl B = [bijJnxp, , = [bji]pxn;therefore BTATis a matrix of orderp x m., m x p., It is obvious that C = AB = [co] is a matrix of order, A T = [ajilnxm and, , Then, , Thus the matrices (AB)Tand BTATare conformable, , because each is a matrix of, , m., of AB, Now (j,i)th element of (AB)T= (i, nth element, , E aikbkjfor all values of i and j, b2j ... bnj and the elements of ith columnof, The elements ofjth row of BT(or i) are bij,, BTAT is, A T (or A) are ail, 42, .. ain), then the (j, i)th element, bkj • aik =, , aik • bkj for all values of i and j., , and their corresponding elementsare, Thus the matrices (AB)Tand BTATare conformable, equal. Hence, , This result is known as the reversal law of transposes., , The Conjugate of a Matrix, obtained from A by replacing, If A is any matrix havin complex numbers, then the matrix, conjugate of maTii¯Räiidäs, its each element by its conjugate complex num er, IS called the, denoted by A or A*., A = [aij], then A * = [an*], Thus, if, where aij* is the complex conjugate of air, For example, if, , 1+2i 2, 3, , , then A* =, , 1-2i 2, 3, , -5i, , 2+3i, , Some properties of the conjugate of a matrix. If A* and B* are the conjugates, , Of, , matrices A and B respectively, then, (b) (A + B)* = A* + B*, A and B being conformable, being any complex number, Cc; (XA*)=, (4) (AB)* = A*B*, A and B being conformable for multiplication., Proof. (a) Let A = (a ijJmxn•Then A* = [a i}*Jmxnwhere aij* is the complex conjugateOfdi/, ], Now (A*)* = complex conjugate of, and B = [aij]mxn, (b) Let

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(A •, , complex conjugate of, complex conjugate of la j, , the conp«ates, Let A, , complex conjugate of 1b I, , the sunt of two matrices is the sum of their conjugates, Then A •, , and, , But, , Orrefore, A, , (d) Let, , A•, , obviousthat (ABr is an, , and B, and, 1b, p matrts and A• is also m x p matnx. then (AB)•, , A•B• are comforrnable., , Now(i.j)th element of A• IF =, , aja, , Also (i. nth element of AB, Therefore(i. nth element of (AR)• = complex conjugate of (li awl., , Sinceexpressions@and (ii) hold for all values of i and j. we must have, , i.e.thecon u ate o the roduct of two matrices is the product. in the same order. of their, con u ales., , v2•S.The Conjugate Transpose (or the Transposed Conjugate) of a Matrix, The conjugate of the transpose ofa matrix A is called the conjugate transpose of A and is, , denotedby At (read as A-dagger) i.e. At a (ATP., , It is obvious that the conjugate of the transpose of a matrix is the same as the transpose of, theconjugateof the matrix i.e. At =, = (At)T., Thus. if A =, then At = (a,ilnxm, For example, if A =, , 1+2i 5i, , 2+9i, , 0, 1, , then, , I-2i, , 1, o, , 2-9i, , 3, , S+3i

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Mathematical, , 136, , PhY&i, , Some properties of the transposed conjugate Qfa matrix,, A and B respectively, then, are the transposed conjugates of matrices, , (At)t, , for addition, _Cb) (A +, = At + Bf, A and B being conformable, e(c) (kA)t = X*At, being any complex number, multiplication., V) (AB)t = Bt At, A and B being conformable for, Proof: —(a) Let A =, Then, conjugate of [aji*Jnxm, Therefore (At )t = transposed conjugate of At = transposed, and B =, (b) Let A = [Clij]mxn, , Then At = [aji nxmand Bt = [bja*]nxm, ij mxn, , Also, , +b, ji nxm= [aji* ji nxm, , so, , ji nxm, , (c) Let A = [aij]mxn.Then At =, , = X*At, Then (LA)t =, and B = [bij]nxp, (d) LetA =, and Bt = [bji*Jpxn, Then At =, It is obvious that C = AB = [co] is a matrix of order m x p., Then (AB) t = [c i)*Jpxm= a matrix of order p X m, , Also, it is obvious that Bt At is matrix of order p x m, , Thus the matrices (AB)t = Bt At are conformable because each is a matrix or order p x m., , Now (j, i)th element of (AB)t = (i, j)th element of, , =, , aik bkj, , a*ik b*kj for all values of i andj., , t, The elements of jth row of Bt are bij*, b2j*, ... bnj* and the elements of ith column ofA, are ail , an, ... am*, then, (j,i)th element of Bt At =, Thus the matrices (A, , a k*. bkj* for all values of i and j., , bkj* ctik*=, , and Bt At are conformable and their corresponding elementsare, , equal. Hence, , (AB)t =, , At, , This result is called the reversal law for conjugate transposes., , 2•9. Symmetric and Antisymmetric Matrices :, A square matrix A = [aij] is said to be symmetric, provided, aij = aji for all values of i and j, If AT is the transpose of a square matrix A = [aij], then, , ji nxn

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137, , Matrices, , Therefore(1) itnplies, , that, , Thus a symmetric matrix is one whose transpose coincides with the matrix itself., (2) represents the necessary and sufficient condition for the matrix A to be symmetric., , For example the matrices b, A square matrix A, , c, , d, , 2, , 4, —5, , —5 are symmetric matrices., 6, , is said to be antisymmetric or skewsymmetric provided, , aij = —aji for all the values of i and j., If ATis the transpose of the square matrix A = [ai)]nxn,then, , ji nxn, Thereforeeqn. (3) implies that, (4), , ——A. Eqn. (4) represents the, Thus a skew-s mmetric matrix A is one for which T —, , necessaryan sufficient condition for a skew-s mmetnc matrix., For i = j, eqn. (3) gives an = —att, 2aii = 0 i.e an = 0, or, matrix are zero., Henceall the ugggug(, Theexamplesof skew-symmetric matrices are, o, , 2, , 3, , -5, , 0, , -2, —b, , Ex.4. Show that every square matrix can be uniquelyexpressedas the sum of symmetric, anda skew-symmatricmatrix., Solution. Let A = [aijl be any square matrix., Then evidently, the matrix A may be expressed as, whereATis the transpose of A, + AT), Q =, SubstitutingP, , (Punjab 1997, Meerut 2000, 1995), , —AT), eqn, (l) may be written as, , Nowtranspose of P —, , andtranspose of Q = Q T, , Thus, , PT = P and QT= ¯, , This proves that P is symmetric and Q is skew-s mmetric. Thus the matrix A may be, expresse as a sum o a symmetric and a skew-symmetricmatrix. We have now to prove that, therepresentation(2) is unique.