Dol Bgyoko (0 FUNDAMENTALS ON ALGEBRA MATRICES AND DETERMINANTS Introducton Expressons of the form P(x o + x + x + + n x n re clled polynomls The coeffcents o,, n re ndependent of x nd the exponents 0,,, n re zero or postve ntegers The hghest nteger exponent n the expresson of P(x s clled the degree of the polynoml P(x The vrle x represents complex or rel numer In the cse of complex vrles, z s often used nsted of x Terms of the type n x n re monomls P(x x hs degree equl to Q(x s polynoml of degree 0 A(x + x - 7x 6 s polynoml of degree 6 f(x x + x / s not polynoml ( / s non nteger exponent, nor The Fundmentl Theorem of Alger (FTA Defnton: the roots or the zeros of polynoml re the vlues of the vrle x for whch the polynoml s equl to zero Let P(x - x, x s the root of ths polynoml Let P(x x -, x - nd x re the roots of ths polynoml Numercl methods re used to fnd the roots of polynomls of degrees greter thn See Numercl Recpes suroutne lrry for smple progrms for clcultng the roots or zeroes of polynoml The Fundmentl Theorem of Alger (FTA: A polynoml of degree n hs exctly n roots (These roots my e rel or complex nd some of them my e equl! The mthemtcl proof of ths theorem cn e found n pproprte lger textooks We prove t n Physcs 4 usng propertes of functons of complex vrle Once ts roots re known, gven polynoml cn e wrtten s product (fctorzton of polynoml If P(x x -, the roots x, x - Then P(x cn e wrtten s P(x (x x (x x (x -(x + Fctor Theorem: f x 0 s root for P(x of degree n, then P(x cn e wrtten s the product of polynoml Q(x of degree (n- nd (x-x 0 : P(x(x-x 0 Q(x Algerc equtons
Dol Bgyoko (0 Algerc equtons nclude those resultng from equtng polynomls x x + 9x 5 0 s n lgerc equton 0x - x 4 5 0x x 4 5 0 s lso n lgerc equton Theses equtons do not nvolve dervtves or ntegrls Fundmentls on Mtrces Defnton of Mtrx: A mtrx s rectngulr (or squre rry of numers or functons tht re clled mtrx elements A + 4 6 s rectngulr mtrx B 4 s squre mtrx In generl, A column( 4 4 4 column(4 Lne or row Let n e the numer of lnes or rows (horzontl n mtrx A nd m tht of column ( vertcl; then A s typcl n x m mtrx Row mtrx: B ( 4 s x 4 row mtrx; x n mtrces re row mtrces
Dol Bgyoko (0 Column mtrx: C s x column mtrx; m x mtrces re column mtrces 6 Squre mtrx: mtrx wth the sme numer of rows s of columns s squre mtrx n x n mtrces re squre Element of mtrx: Element s n row nd n column zero or null mtrces re those for whch ll the elements re zero A 0,, 0 Unt mtrx: A unt mtrx, whch s lwys squre for our purposes, s one n whch s zero for nd s for Hence,, δ, I, I 0 0, nd 0 0 I 0 0 re ll unt mtrces 0 0 If A s unt mtrx, then,, δ Kronecker Delt or Kronecker symol : δ 0 for nd δ for So, δ δ δ δ whle δ δ δ δ δ 0 44 4 Dgonl elements: The dgonl elements of squre mtrx A re the elements for whch the row nd the column ndces re equl The dgonl elements re nd Dgonl mtrx: A mtrx D s dgonl or s n ts dgonl form f nd only f for ll elements d, d c+ δ, where the c + re numers lso known s the egenvlues of the mtrx
Dol Bgyoko (0 0 0 D 0 0 hs egenvlues of, -, nd 0 0 Trce of squre mtrx: The trce of squre mtrx s the sum of ts dgonl elements Tr(D l+ (- + 0 Tr(A + for A Trnspose of mtrx: The trnspose of mtrx A s mtrx noted A T or A ~ The trnspose A T T of A s such tht,,, The rows of A re the correspondng columns of A T 4 7 T Let A 4 5 6, then A 5 8 7 8 9 6 9 Note well tht the dgonl elements of A nd of A T re the sme : Tr(A Tr(A T Also note tht the nth column of A T s the nth row (or lne of A Trnsposes of mtrces re wdely used n computtons, quntum mechncs, nd trnsformtons Is the trnspose of row mtrx column mtrx? NOTATION: Mtrx A s lso noted ( A ( : Note well tht, wthout the prentheses, refers to specfc mtrx element ε k s the Lev-Cvt symol or densty ε k f,, k, re ll dfferent nd n the order of - f,, k re ll dfferent nd NOT n the order 0 f ny two ndces re equl (, k, or k Even numers of permuttons of -- n tht order gve : ε ε ε ; Odd numers of permutton gve -: ε ε ε The sgn of ε k s chnged y ny sngle permutton of consecutve ndces: ε ε ε ε k k, k k 4
Dol Bgyoko (0 Note tht n even numer of permuttons does not chnge the sgn A permutton of two ndces tht re not consecutve my represent n even or n odd numer of permuttons of consecutve ndces, dependng on how mny ndces re etween the two eng permuted! Determnnts of Mtrces Let A, the determnnt of A, noted A det A If A, then det(a Note the rs- for determnnts- nd prentheses for mtrces + + + det A + + ( ( ( ( ( ( The ove determnnt ws clculted (developed usng the frst row Theorem: A determnnt cn e evluted long ny row or ny column Ths theorem s extremely useful n the prctcl evluton of determnnts From t, t redly follows tht det(a det(a T A useful defnton follows from the ove expresson of det (A By defnton, the cofctor of the element of mtrx s the expresson tht s multpled y when det(a s developed long row or column tht ncludes From the ove expresson of det(a, the cofctor of s ( ( s ( + ( 4 Opertons on Mtrces + nd tht of Equlty: Two mtrces A nd B re equl f nd only f,, Ths defnton hs trvl consequences: n m n nd n (m ± n mtrces cnnot e equl; two squre mtrces of dfferent dmenson (m m nd n n wth m n cnnot e equl 5
Dol Bgyoko (0 Addton: Addton s defned etween mtrces s follows: C A + B,, C + Ths defnton mples tht only m n mtrces cn e dded n the ordnry sense of ddton For squre mtrces, the numer of rows or of columns s clled the dmenson of the mtrces Only squre mtrces of equl dmensons cn e dded A B C ( + 0 C ( (0 + (0 + ( + ( ( + ( + (7 + 5 Multplcton of Mtrces The product of two mtrces A nd B s mtrx C such tht C k kk, where k to N C s otned y multplyng, term y term, the elements n lne of A y the elements n column of B Exmple: A B c AB c c c ( ( + + ( ( + + Notes: For the product of mtrces to e defned, the numer of columns of the frst mtrx must e equl to the numer of rows of the second mtrx Tke some smple mtrces to see tht n generl AB BA: the multplcton of mtrces s not commuttve The neutrl element, for mtrx multplcton, s unt mtrx I n ; for mtrces, 6
Dol Bgyoko (0 I 0 0 ; for x mtrces I 0 0 0 0 0 0 Prove tht the set of n n mtrces consttutes n Aeln group when endowed wth the nry operton of ddton s defned ove (Do not confuse provng wth verfyng n prtculr cse Provng estlshes the vercty of proposton for ll cses! 6 Inverse of Mtrx By defnton, the nverse of mtrx A s mtrx noted A - such tht AA - A - A I, where I s the unt mtrx Fnd n your textook nd lern some methods of clcultng the nverse of mtrx See, for nstnce, pge If the nverse of squre mtrx exsts, then one of the wys to clculte t s s follows A T C det A where C s mtrx whose elements re the cofctors of elements of A: c cofctor of A mtrx whose determnnt s zero s sngulr mtrx nd t does not hve n nverse WARNING: For lrge mtrces, 0x0, 00x00, etc, the clculton of the nverse s done usng computer It s extremely mportnt to rememer, however, tht smll or tny round-off errors n the clculton of the nverse of mtrx led to huge errors n the fnl result Consequently, these clculton must e done usng s hgh precson s possle f one s to vod spurous results for the nverse (hgh precson mens tht every clculton s done y crryng s mny decml plces s possle 7 Drect Product of Two ( Mtrces By defnton, the drect product of two mtrces A nd B s mtrx D otned y multplyng every element of A y mtrx B Note: f s memer, to multply or dvde mtrx y mens to multply or dvde every element of the mtrx y Followng the ove defnton of the drect product, we get, for exmple, 7
Dol Bgyoko (0 A B nd A B Trvlly enough, the drect product of two mtrces s 4 4 mtrx The drect product of mtrces does not stsfy closure property, e, the drect product of two mtrces s not mtrx Note: The symol, etween two mtrces, sgnfes drect product nd not smple multplcton or product, unless otherwse ndcted explctly Some textooks use to further vod confuson Let us thnk t nd note tht someone who does not know thng out the drect product my nterpret to men smple product! Applcton: The drect product s extensvely used n some res of quntum mechncs (the theory of the moton nd propertes of electrons nd other prtcles whle they re nsde toms, molecules, solds, nd lquds Ordnry opertons on mtrces (ddton, multplcton, trnsposton, nverse clculton, etc re extensvely used n most res of most scence nd engneerng dscplnes, nd n socl scences 8
Dol Bgyoko (0 8 Specl Mtrces Before defnng some key specl mtrces, we recll some sc notons nd nottons: I s unt mtrx, I δ * * A or A s the complex conugte of A,, or s the complex conugte of (The complex conugte of complex numer s otned y chngng the sgn of the pure mgnry numer or throughout ( to - nd - to ~ ' A T T or A or A s the trnspose of A,, (lnes or rows of A ecome the correspondng columns of A T B s the Hermtn conugte of A T B ( ~ * A A * A B s the dont or dugte of A B Trnspose of the mtrx formed y cofctors of elements of A The dugte of A s often noted Ad (A or  The nverse of A s A - Ad( A A det( A A generl complex numer z + where nd nd re rel numers If z + 0, then z s rel (e, s rel numer If z 0 +, 0, then z s complex numer often clled "pure" mgnry numer For 0 nd 0, z s generl complex numer 7 Tle of Specl Mtrces A s rel *, s rel numer (e, A s complex t lest one element of A tht s complex (e, such * tht A s symmetrc A A T or, A s skew-symmetrc or A -A T, nt-symmetrc (Note: then the re zero nd the trce s zero! A s orthogonl A T A - or AA T A T A I, the unt mtrx U s untry UU U U I (or U U - H s Hermtn H H T* or h h *,, (the egenvlues of Hermtn mtrx re rel numers 9
Dol Bgyoko (0 Plese see numercl nlyss ooks (Numercl Recpes s n the Physcs Redng Room for mny other specl mtrces Assgnment: Prove tht the set of m x m mtrces, for m fxed, n whch the nry operton of ddton s defned s done ove, consttutes n Aeln group The opposte of mtrx A s -A, such tht A + (-A (-A +A 0 mtrx Sutrcton s smply prt of the operton of ddton To sutrct mtrx A s to dd ts opposte The opposte of mtrx A s A: A + (-A mtrx 0 0 0 0 If A, -A, nd A + (-A ϕ DIAGONALIZATION OF A MATRIX Equtons of the form A re egenvlue equtons, Lmd, multplyng, s numer A s mtrx In generl, nd re t lest un-column mtrces The generl form of egenvlue equtons s: 0 0 [mtrx (or opertor] [functon ] [numer] [functon ] If A s mtrx, then lso s mtrx such tht the multplcton of the two mtrces s possle So, f A s mtrx, then could e mtrx s shown n exmples dscussed elsewhere [See your textook for detls nd llustrtons] To solve the ove egenvlue equton s the sme s to dgonlze mtrx A There re severl wys for dgonlzng mtrx The one we must know follows (see the textook for others One dgonlzes mtrx A y settng the determnnt of ( A I equl to zero nd y solvng the resultng equton: (for nxn mtrces, ths determnnt wll e polynoml of degree n! A I 0 The soluton of ths equton re the egenvlues of A 0
Dol Bgyoko (0 0 0 0 0 0 0 Then clculte the determnnt of ths lst mtrx; t s polynoml of degree Hence, ccordng to the fundmentl theorem of lger, ths polynoml hs roots (An n n mtrx, y vrtue of the FTA, wll hve n egenvlues Let us dgonlze the followng mtrx (e, solve the followng egenvlue equton To solve ths egenvlue equton s to dgonlze the mtrx ( 0 or ( 4 0 Hence, ( 4 0 nd ( ± or + nd Consequently, nd For ech egenvlue of the mtrx there corresponds n egenfuncton Wth the ove egenvlues, let us fnd the correspondng egenfunctons We fnd y solvng the equton A where A nd re gven ove ( ( ( 0 0 0 0 So, + 0 nd + 0 A soluton of ths system of two equtons wth two unknowns s We cn tke tht constnt to e Hence, We cn verfy tht Proceedng smlrly, we fnd elow the egenfuncton correspondng to A Set Then, A ecomes:
Dol Bgyoko (0 A Ths leds to 0 0 0 + + nd or Therefore, stsfes ths system of equton The rtrry vlue cn e set to e equl to Consequently, We esly verfy tht For ech egenvlue of mtrx, there s n egenfuncton [These egenfunctons re column mtrces n our exmple ove] A ; or s sd to e normlzed when ts squre mgntude s ( * + Ths squre mgntude s not nd hence s not normlzed To normlze, let us defne new functon tht s dvded y the squre-root of ts squre-mgntude of : Set φ Ths new functon, φ, s the normlzed egenfuncton correspondng to It s esly verfed tht * φ φ Hence, φ s normlzed Proceedng smlrly, we cn fnd the normlzed the egenfuncton correspondng to
Dol Bgyoko (0 Some key ponts: Dgonlzton tkes mtrx nd trnsforms t nto ts dgonl form Mtrx Dgonlzton, see the textook, cn e ccomplshed usng untry trnsformtons! Theorem- the dgonlzton process does not chnge the trce of mtrx 0 A 0 Tr( A Tr( dgonla Egenvlues of mtrces re fundmentl physcl propertes If the mtrx s tht of the moment of nert, the egenvlues wll e the moments of nert for rottons round the prncpl xes! In sc quntum mechncs, the egenvlues resultng from the Schrödnger equton represent energes of prtcles Crtclly Importnt Pont Aout Mtrx Inverson nd Dgonlzton (Need for hgh precson rthmetc n mtrx nverson or dgonlzton For most mtrces, the clculton of the nverse s done usng computer The sme s true for the dgonlzton of mtrces These two opertons, however, s extremely senstve to errors: Specfclly, smll roundng errors n the nverson of mtrx cn led to huge errors n the nverse mtrx So, mtrx nverson, when performed usng computer, must e done usng doule, qudruple, or hgher precson In computer progrmmng rgon, sngle precson rthmetc crres dgts up to the sxth decml plce whle doule precson rthmetc crres dgts up to the th decml plce When workng on mportnt ssues (medcl reserch, trectory of spcecrft, energes levels of quntum prtcles, etc one MUST PAY EXTREME ATTENTION TO GUARANTEE THAT NO SPURIOUS RESULTS ARE OBTAINED BY CARELESSNESS (flure to utlze hgh precson rthmetc, e, doule, qudruple, etc, precson In quntum mechncs, you wll encounter egenfunctons of the followng type, whch cn not e normlzed utlzng the ove process In such cse, utlze the method of ntegrton to normlze (r Thus, nsert the egenfuncton, (r, nd ts complex conugte nto the followng equton: r ( r e, * ψ ( r ψ ( r dv N, where N represents the vlue of the ntegrl If N, then the functon s normlzed; however f N, you must construct new functon, dvded y the squre-root of the vlue of the ntegrl, N ψ ( r φ ( r, whch s N
Dol Bgyoko (0 4 SYSTEM OF ALGEBRIAC EQUATIONS As prevously mentoned n Secton, lgerc equtons do not nvolve dervtves or ntegrls, nd the system cn hve the followng form: x + y + z c x + y + z c x + y + z c It s mportnt to note tht you must lwys plce the system n queston nto the ove cnoncl form A system of lgerc equtons cn e homogeneous or nhomogeneous A System of Homogeneous Algerc Equtons The system s homogeneous f, nd only f, every term n the equtons whch, does not contn vrle, s zero Thus, c,c, nd c s gven ove must e zero A System of Inhomogeneous Algerc Equtons An lgerc equton s nhomogeneous f you hve term tht does not contn vrle nd s not zero Note tht only one equton needs to e nhomogeneous for the system to e nhomogeneous THEOREMS (A system of homogeneous lgerc equtons hs non-trvl solutons f nd only f the determnnt of the mtrx of the coeffcents s zero 0, If the mtrx s not equl to zero, then non-trvl soluton does not exst (A system of nhomogeneous lgerc equtons hs non-trvl solutons f nd only f the determnnt of the mtrx of the coeffcents s dfferent from zero 0 4
Dol Bgyoko (0 5 CRAMER METHOD (INHOMOGENEOUS ALGEBRAIC EQUATIONS The Crmer Method s utlzed to solve system of nhomogenous lgerc equtons, e, c,c, nd c re not ll equl to zero (e, t lest one s non-zero Let system of nhomogeneous, lgerc equtons e gven s follows: x + y + z c x + y + z c x + y + z c Let c, c, nd c Let C e the mtrx whose elements re the coeffcents s shown elow C If the determnnt of C s not equl to zero, then one clcultes C x, C y, nd C z s follows you replce ech respectve column of the mtrx y c,c, nd c to fnd the determnnt of ech mtrx Exmple: C x, det (C x? to clculte; C y, det (C y? to clculte; nd C z, det (C z? to clculte Then, y the Crmer Method (lso clled Crmer rule, we hve x det ( c x, y det ( c det ( c y, nd z det ( c det ( c z det ( c 5
Dol Bgyoko (0 6 THEOREMS ON LIMITS OF POLYNOMIALS THEOREMS ( A polynoml s functon of ny vrle such s x, t, or n goes to the term wth the hghest exponent when the vrle x, t, or n, goes to nfnty ( As such, the lmt of the polynoml s the term wth the hghest exponent when the vrle goes to nfnty ( Exmple: P(t t + t 7 9t 8-9t 8 when t ( A polynoml s functon of ny vrle such s x, t, or n goes to the term wth the lowest (smllest exponent when the vrle x, t, or n, goes to zero As such, the lmt of the polynoml s the term wth the smllest exponent when the vrle s gong to zero Exmple: P(t t + t 7 9t 8 t when t 0 Applctons of the Theorem ( ove to the study of the convergence of nfnte seres n 5n4 + 9n5 9n5 9 U ( n n, for n So ths seres does not even meet + 5 5n n n 5n the necessry condton for convergence (s we wll see n the next lesson Hence, t dverges 6