Matrix Algebra CHAPTER 1 PREAMBLE 1.1 MATRIX ALGEBRA

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1 CHAPTER 1 Mtrix Algebr PREAMBLE Tody, the importnce of mtrix lgebr is of utmost importnce in the field of physics nd engineering in more thn one wy, wheres before 1925, the mtrices were rrely used by the physicists. In mny engineering opertions, we need to invoke the concept of mtrix, nd sometimes, it is more of routine work wherein the physicl quntities re expressed by liner opertors on vector spce. Very often, there is quite close correspondence between experimentl vlues with those clculted by the use of this forml or rther bstrct method, which nturlly gives lot of credibility to mtrix formlism. Hence, cler concept of mtrix lgebr should be given in the first chpter of this book. In quntum mechnics nd in mny other modern brnches of physics, we hve to operte through mtrix, nd the mtrix representtion is more often necessry, i.e., how Heisenberg Mtrix formlism cme into existence. 1.1 MATRIX ALGEBRA The study of mtrices is mostly motivted by the necessity to solve system of liner simultneous equtions of the form: 11 x x n x n = b 1 21 x x n x n = b 2 n1 x 1 + n2 x nn x n = b n (1.1) where, x 1, x 2,.... x n re the unknowns. The eqution (1.1) cn be esily expressed in the mtrix form s: Ax = b (1.2) where, A is squre mtrix of dimension (n n). Here, x nd b re vectors of dimension (n 1). All these quntities re given by the following reltions: é n éx1 éb1 A= n, x = x 2, b = b ën1 n2... nn ëx n ëb n From the bove mtrices, we notice tht mtrix is simply n rry of elements, clled mtrix elements in the usul prlnce in the field of physics nd engineering. The bove

2 2 Mthemticl Physics for Engineers mtrix A cn lso be denoted s [A]. There re different elements in different positions of the rows nd columns, e.g., n element locted t the i th row nd j th column of the mtrix A is simply denoted by ij. If ij = 0 for ll i nd j, then A is clled null mtrix. The mtrix is lso denoted by [ ij ]. The question of multipliction of two mtrices A nd x is implicit in the bove equtions. The dot product of the i th row of A with the vector x is equl to b i giving rise to i th eqution in (1.1), s lso shown little lter in this section Row nd Column Vectors A mtrix of dimension (1 n) is clled row vector, wheres mtrix of dimension (m 1) is clled column vector. Let us tke some exmples s: 1. c = [1 1 3] is (1 3) row vector, nd é d = is (4 1) column vector. 4 ë MATRIX OPERATIONS After properly defining the bsics of mtrix lgebr, it is useful to describe importnt opertions for mtrices s below: Addition nd Subtrction Let us consider two mtrices A nd B with the dimensions of both (m n). Then, the sum C = A + B is defined s: c ij = ij + b ij (1.3) It mens tht the (ij) th component of C cn be obtined by dding the (ij) th component of A to the (ij) th component of B. é 3 2 é1 2 + ë 2 4 ë4 0 = é 4 0 (1.4) ë2 4 Similrly, the subtrction of the mtrices cn be defined nd explined. The following lws re lso vlid for ddition of the mtrices of the sme order in two different cses s: () For Commuttive Cse: A + B = B + A (b) For Associtive Cse: (A + B) + C = A + (B + C) Multipliction by Sclr The multipliction of mtrix A by sclr c cn be defined s: ca = [c ij ] (1.5) As n exmple, let us write the following mtrix s: é ë = 3 é ë 5 7 (1.6)

3 1.2.3 Mtrix Multipliction Mtrix Algebr 3 Let us tke (m n) mtrix A nd nother mtrix B of dimension (n p). The product of these two mtrices results in mtrix C of dimension (m p), s shown below: A B = C (1.7) (m n)(n p) (m p) The (ij) th component of C is obtined by tking the dot product s: c ij = (i th row of A). ( j th column of B) (1.8) As n exmple, let us show tht: é1 4 é ë = é (1.9) ë 0 3 ë 10 7 It hs to be noted tht AB ¹ BA. Actully, BA my not even be defined, since the number of columns of B my not equl the number of rows of A. The commuttive lw of multipliction is not vlid (in generl) for mtrix product, i.e., AB ¹ BA. But, the ssocitive lw of multipliction is vlid for the mtrix product, i.e., A(BC) = (AB)C Differentition nd Integrtion It is known tht the components of mtrix do not hve to be sclrs. They my lso be functions. As n exmple, let us tke: A= é 2 x + y x 2xy (1.10) ë4 + x y In such sitution, the mtrix cn be differentited nd integrted. The derivtive or the integrl of given mtrix is simply the derivtive or the integrl of ech component of the mtrix. Therefore, we cn write these mtrices s: d édij () x A( x) = (1.11) dx ë dx ò Adxdy = é ëò ij dxdy (1.12) If A be the mtrix of constnts hving dimension (n n) nd x = [x 1, x 2,.... x n ] be column vector of n vribles, then we cn cite n interesting cse by using eqution (1.12), whereby the derivtive of Ax with respect to the vrible x p is given s: d dx p (A x) = p (1.13) Here, p is the p th column of the mtrix A, which cn be esily verified by writing down the whole eqution Ax in the mtrix form. 1.3 PROPERTIES For n rbitrry mtrix A, it is quite importnt to know some bsics of their properties like trnsposition, complex conjugtion, etc. Now, some of these importnt properties of the rbitrry mtrices will be discussed here.

4 4 Mthemticl Physics for Engineers Trnsposition If the mtrix A = [ ij ], then the trnspose of n rbitrry mtrix A, usully denoted by A T, is defined by A T = [ ji ]. Therefore, the rows of A becomes the columns of A T. Let us tke n exmple s: é A = then, A T é = 0 5 ë ë 3 2 Generlly speking, if A is of dimension (m n), then A T hs to be of dimension (n m), i.e., the trnsposition is chieved by interchnging corresponding rows nd columns of A. The trnspose of product is defined s the product of the trnsposes in reverse order s: (ABC) T = C T B T A T (1.14) Complex Conjugte Mtrix The complex conjugte of n rbitrry mtrix A is formulted by tking the complex conjugte of ech element. Let us tke n exmple s: A* = ij * for ll i nd j. é3 + 2i 4 6i * é3 2i 4 + 6i A=,A = ë 4 3i ë 4 3i If A* = A, then A is rel mtrix, which is importnt in physicl situtions. In this ctegory, there is nother importnt mtrix clled Hermitin conjugte, denoted by Ay, which is obtined by tking the complex conjugte of the mtrix, nd then the trnspose of this complex conjugte mtrix, which is useful in certin pplictions. 1.4 SQUARE MATRICES The squre mtrix is so simple tht it is not discussed here with ny detils, since we mostly come cross such mtrices in the engineering problems. Any mtrix whose number of columns equls the number of rows is clled squre mtrix, e.g., simple (2 2) mtrix or (4 4) mtrix commonly occurring routinely in our problems. Certin importnt squre mtrices re described below Digonl Mtrix After understnding simple squre mtrix, it is esy to grsp digonl mtrix, which is ctully squre mtrix with nonzero mtrix elements only long the principl digonl. This type of mtrix is commonly encountered in mny problems of solid stte physics. The typicl exmple of digonl mtrix is shown s: é3 0 0 A= (1.15) ë0 0 6 Here, it is clerly seen tht the nondigonl mtrix elements re ll zero, wheres the digonl elements re ll finite numbers. It hs to be noted tht if the vlue of the determinnt of the mtrix A, i.e., det A = 0, then this mtrix is sid to be singulr mtrix.

5 1.4.2 Identity Mtrix Mtrix Algebr 5 This is lso clled unit mtrix. This identity or unit mtrix is digonl mtrix with 1 s long the principl digonl, s shown below: é I= (1.16) ë If I is of dimension (n n) nd x is vector of dimension (n 1), then we cn write: Ix = x (1.17) This unit mtrix cn be generlly described by Kronecker delt (d ij ) Symmetric Mtrix A symmetric mtrix is gin squre mtrix, whose mtrix elements stisfy the following: ij = ji (1.18) or, on n equivlent bsis, for A to be symmetric mtrix, we cn write it s: A= A T (1.19) It mens tht ll the mtrix elements, which re locted symmetriclly with respect to both sides of the principl digonl, re equl, s evident below: é A= (1.20) ë If A T = A, then A is clled n ntisymmetric or skew mtrix. These mtrices hve importnt pplictions s Puli mtrices to describe the spin properties of n electron. Typiclly, this cn be shown s: é0 i T é 0 i x 2 =, x2 = ëi 0 i 0 = x 2 ë Upper Tringulr Mtrix This is simple mtrix, but it is quite uncommon in engineering problems. In this cse, it is mtrix whose mtrix elements below the principl digonl re ll zero, s show below: UT = é ë (1.21) Determinnt of Mtrix Here, gin we del with squre mtrix. The determinnt of squre mtrix A is sclr quntity, which is denoted by det A. There is method of cofctors, which re used here to show the determinnts of (2 2) nd (3 3) mtrix s:

6 6 Mthemticl Physics for Engineers é det ë é det ë = (1.22) = ( ) ( ) + ( )(1.23) Mtrix Inversion This is perhps one of the most importnt of ll mtrices discussed so fr due to its sheer importnce in the pplictions of mtrix lgebr in the field of engineering nd mthemticl physics. Here, gin we del with squre mtrix A. Now, if det A ¹ 0 (to void infinity ), then the mtrix A hs n inverse, which is simply denoted by A 1. The inverse mtrix stisfies the following reltion s: A 1 A = A A 1 = I (1.24) If the det A ¹ 0, then the question of nonsingulrity comes, nd we cn sy tht the mtrix A is nonsingulr. But, if det A = 0, then we get singulr mtrix A, nd in this cse the inverse cnnot be defined for obvious resons. By eliminting the ith row nd the jth column of squre mtrix A, we get the minor M ij, which is the determinnt of (n 1 n 1) mtrix. Here, the cofctors C ij of mtrix A cn be written s: C ij = ( 1) i+j M ij (1.25) Adjoint of Mtrix The mtrix elements C ij mkes the mtrix C, which is clled cofctor mtrix, which hs reltion with the djoint of mtrix. The djoint of this mtrix A cn be defined s: Adj A = C T (1.26) But, this djoint mtrix A hs lso reltion with the inverse of squre mtrix A, which cn be described s: A 1 = Adj A / det A (1.27) As n exmple, we cn write the inverse of (2 2) mtrix A s: é ë = 1 é det A ë (1.28) This mtrix long with selfdjoint mtrix is very importnt in quntum mechnics nd in other fields of mthemticl physics. If Adj A = A, then A is sid to be selfdjoint. Here, it is importnt to mention bout nother importnt mtrix in physics, i.e., Hermitin mtrix, i.e., Ay = A, then A is sid to be Hermitin mtrix, which is lwys rel, which is necessry nd lso useful in quntum mechnics to find the vlue of different mesurble vlues of the observbles. The other importnt squre mtrices like Orthogonl mtrix re not discussed here, since the entire gmut of squre mtrices described bove lredy give enough insight into different types of useful mtrices in mthemticl physics.

7 Mtrix Algebr EIGENVALUES AND EIGENVECTORS This is the most importnt topic in the mtrix lgebr. It is very useful in quntum mechnics nd host of other subjects in physics nd engineering. First of ll, we hve to pose problem s n eigenvlue problem. Let us consider the eigenvlue problem s: Ay = ly (1.29) where, A is the usul squre mtrix (n n) signifying liner opertor, s described bove, y is n eigenvector or eigenfunction nd l is the chrcteristic or eigenvlue. Here, if we desire nontrivil solution, i.e. we wnt nonzero eigenvector y nd the consequent eigenvlue l, which must stisfy the bove eqution (1.29). We cn lso write the mtrix form of n eigenvlue problem by using eqution (1.29) s: (A li)y = 0 (1.30) It is very esily noted tht nonzero solution for y will be obtined when (A li) is singulr mtrix, or it cn be rticulted s: det (A li) = 0 (1.31) This eqution (1.31) is normlly clled the seculr or chrcteristic eqution of A. This eqution cn be solved for the n roots or rther different eigenvlues l 1, l 2,....l n. For ech of these eigenvlues (l i ) obtined by expnding the determinnt in eqution (1.31), the corresponding eigenvectors (y i ) cn then be obtined s: (A l i I)y i = 0 (1.32) It hs to be noted tht the bove eigenvectors (y i ) cn be determined only to within multiplictive constnt, since (A l i I) is singulr mtrix. It is better to tke n exmple to explin the eigenvlue problem s: Exmple 1 A= é 3 4 ë4 3 In order to mke n eigenvlue problem, the bove mtrix cn be written in the usul determinnt form s: 3 l l = 0 or, 9 + l 2 16 = 0 l = ± 5, l 1 = 5, l 2 = + 5 So, we get the eigenvlues (l 1 nd l 2 ). Now, we hve to find the corresponding eigenvectors s: é1 x 1 = ë 2 nd, x 2 = é 2 ë1 Let us tke é1 2 P= ë 2 1 Then, we cn write the inverse of this mtrix s:

8 8 Mthemticl Physics for Engineers P 1 1 é 1 2 = 5 ë 2 1 Therefore, finlly, we cn write it s: P 1 1 é 1 2é3 4é AP = ë 2 1ë4 3ë 2 1 = 1 é 1 2 é ë 2 1 ë é25 0 é5 0 = = 5 ë 0 25 ë 0 5 = é l1 0 ë 0 l2 The bove sums up the eigenvlue problem, which hs myrids of useful exmples (s described briefly in the Premble), in different brnches of physics nd engineering.