Handout #2. Introduction to Matrix: Matrix operations & Geometric meaning

Transcription

1 Hdout # Title: FAE Course: Eco 8/ Sprig/5 Istructor: Dr I-Mig Chiu Itroductio to Mtrix: Mtrix opertios & Geometric meig Mtrix: rectgulr rry of umers eclosed i pretheses or squre rckets It is covetiolly deoted y cpitl letter Mtrix is powerful tool to orgize dt A lot of sttisticl methods ivolve the mipultio ie, trsformtio of dt mtrix, the elemet i mtrix A is deoted y i, ; the suscript i i th row d th colum re idices tht tell the loctio of elemet i, The lrgest umer of i d tells the dimesio order of mtrix For exmple,, = d mtrix A is of order x red s two y two Cosider other mtrix B, where the elemet is deoted y i, Sice the lrgest i = d lrgest =, the mtrix B is of order x B = Q: wht is the elemet of, d,? c =, d =, e = symol x, with rrow, is ofte used i mth clss Mtrix c is of order x, d is of order x d e is of order x Whe there is oly oe row colum i mtrix, it s termed s row colum vector We c show the vector c & d i the X-Y ple for e, three dimesiol spce is eeded d -, Y Suppose c = [c, c,, c ], the legth orm of c is c = c c c O c, X Q: Plese fid d i the left grph

2 Alger of Mtrices Equlity of mtrices If B, the i, = i, Exmple:, B = Sclr multiplictio k i, k Exmple: k =, k = 9 c Additio d Sutrctio A B = i,, i Exmple: + =, - = d Mtrix multiplictio A m x d B x p must e coformle AB = m m p p p = p m eg = = X Y Z,,,,,, e=,, Vector e i three-dimesiol spce X, Y, Z e =,, d its orm e = =

3 eg Numer of crs Numer of uses Mody 5 Tuesdy 5 5 Wedesdy 5 5 Price = $/cr, $8/us, fid the reveue R o Mody, Tuesdy, d Wedesdy 5 R = 5 5 = Additiol exmple q = 5 7, z =, P = 5, w = 8 Q: Plese show the mout of profit i terms of mtrix opertio eg x + x = x x = - Ax =, where x, x =, = x AB BA, uless B, B = We c get the product of AB ut ot BA e Rk of mtrix: mximum umer of idepedet rows or colums f Trspose of mtrix symol: A T or A, B =

4 A T =, B T = Rules: A = A A B = A B AB = B A g Specil mtrices g: if the umer of rows equls the umer of colums, it is squre mtrix g: A is squre mtrix, d if A, the A is symmetric mtrix eg, Notice: For symmetric mtrix A, i, =, i i g: Digol mtrix ll the off-digol elemets re zero eg, C = g: Idetity mtrix ll the digol elemets re oe d off-digol elemets re zero eg, I = g5: A squre mtrix A is idempotet if A = A = eg,, A, AA still A do ot chge

5 h The trce of squre mtrix A x trce + + the sum of ll the digol elemets eg, the trce of the previous idempotet mtrix is oe = / + / + / Q: Wht is the trce of x idetity mtrix? i Determit of squre mtrix det A or show geometric meig Y D -, Suppose O C, X det = Suppose A is x mtrix, where, how do we fid det A Cofctor Expsio pproch: Step : choose y colum or row Step : fid the mior determit of su-mtrix of ech elemet Step : fid the cofctor Step : multiply ech elemet y its cofctor d get the sum of these products Step : choose the first row Step : Mior: A = det = det 5

6 A = det = det Step : Cofctor C i, = - i + A i, C = - + A Step : det i, C i, or i, C i, if expded y colum i eg, = A = det = 5, C = - + A = 5 A = det = -, C = - + A = A = det = -, C = - + A = - det = Iversio of squre mtrix A - Defiitio: A - I eg,, let A - = A - =, four equtios d four ukows; we eed to solve simulteous equtio system i order to fid ll the elemets i A - How do we fid the iverse of squre mtrix A x if?

7 Formul: A - = det A d A, where d A is termed doit mtrix C C C d C C C T = C C C Cofctor C i, = - i + A i, C C C C C C C C C eg, det -5, d A - = det A Doule check: d - 5 = AA - = = 5 Q: fid the iverse of the followig x mtrix B B -? B = Rules: A - AA - = I A - - = A AB - = B - A - A - = A - 5 det A - = det A 7

8 Solve Simulteous Lier Equtios x + x = x + x = Let x, x =, = x Ax = A - Ax = A - Ix = A - x = A - we solve x usig iversio pproch Erlier exmple: x + x = x x = - x, x =, = x x = = 5 5 eg Supply d demd model Q = P Q = + P P, x =, = Q P x = = A - / / = Q / / = 8

9 k Eigevlue Prolem Eigevlue is lso clled ltet vlue or chrcteristic root Aq = q A is kow x symmetric mtrix, is ukow sclr d q is ukow x colum vector The solutio of fidig the ukow sclr Eigevlue d the ukow q Eigevector is clled the Eigevlue prolem eg A - Iq = For otrivil solutio q, A - I x mtrix must e sigulr It mes tht det A - I = A - I = det A - I = = solve for d oti = d = - The umer of o-zero eigevlues c e used to fid the rk of the mtrix I this exmple, there re two o-zero eigevlues Therefore, the mtrix A hs full rk ie, rk or it mes there re two lier idepedet rows colums i mtrix A Next we hve to fid the eigevector: Whe = A - Iq = -q + q = q = / q, where q = [q, q ] T Use the ormliztio coditio: q + q = Solve for q d q we c oti: q = q = 5 5 Eigevector q = 5 correspodig to = 5 Whe = - 9

10 A - Iq = q + q = q = -q, where q = [q, q ] T Use the ormliztio coditio: q + q = Solve for q d q : q =, q = 5 5 Eigevector q = 5 correspodig to = - 5 Property of q d q : q T T q =, q q = Collect q d q i mtrix Q Where Q = [q, q ] Properties of Q: Q T Q = QQ T = I from previous property, Q T = Q - Q is Orthogol mtrix Q T AQ =, where = Property ove is clled Digoliztio c If A is ot symmetric the Q - AQ = Q is o loger Orthogol d trce trce e det i = Coect the ove results d we hve the followig coclusios: If x mtrix A is osigulr det A iff A - exists c rk full rk Note: iff : if d oly if; equivlet ecessry d sufficiet sttemet symol