Section 2.2. Matrix Multiplication

Transcription

1 Mtri Alger Mtri Multiplitio Setio.. Mtri Multiplitio Mtri multiplitio is little more omplite th mtri itio or slr multiplitio. If A is the prout A of A is the ompute s follow: m mtri, the is k mtri, 9 m k mtri whose ( i, j) - etry is Multiply eh etry of row i of A y the orrespoig etry of olum j of, the result This is lle the ot prout of row i of A olum j of. Emple Compute the (, ) (, ) - etries of A where: A The ompute A. Solutio The (, ) - etry of A is the ot prout of row of A olum of, ompute y multiplyig orrespoig etries ig the result:. ( ).. Similrly, the (, ) etry of A is the ot prout of row of A olum of, ompute y multiplyig orrespoig etries ig the result:... Sie A is is, the prout is mtri: Lier Alger I

2 Mtri Alger Mtri Multiplitio Computig the ( i, j) - etry of A ivolves goig ross row i of A ow olum j of, multiplyig orrespoig etries, ig the results. This requires tht the rows of A the olums of e the sme legth. The followig rule is useful wy to rememer whe the prout of A e forme wht the size of the prout mtri is. Rule Suppose A hve sizes m ' p, respetively: m ' p The prout A e forme oly whe mtri A is of size m ' ; i this se, the prout p. Whe this hppes, the prout A is efie. Emple If A, Compute A A, A,, whe they re efie. Solutio Here, A is mtri is mtri, so A, re ot efie. The A A re efie, these re mtries, respetively: A... Lier Alger I

3 Mtri Alger Mtri Multiplitio A= = Ulike umeril multiplitio, mtri prouts equl. A A ee ot e The umer plys eutrl role i umeril multiplitio i the sese tht.. for ll umer. A logous role for mtri multiplitio is plye y squre mtries of the followig types:,, so o. I geerl, ietity mtri I is squre mtri with s o the mi igol zeros elsewhere. If it is importt to stress the size of ietity mtri, eote y I. Ietity mtri ply eutrl role with respet to mtri multiplitio i the sese tht : AI A I wheever the prout re efie More formlly, give the efiitio of mtri multiplitio s follow: If A ij is m is p the i th row of A the j th olum of re, respetively, i i i j j j Hee, the ( i, j) -etry of the prout mtri A is the ot prout: i j i j ij ikkj k Lier Alger I

4 Mtri Alger Mtri Multiplitio This is useful i verifyig ft out mtri multiplitio Theorem Assume tht k is ritrry slr tht A, C re mtries of sizes suh tht the iite opertios e performe:. IA A, I. A ( C) ( A) C. A ( C) A AC; A( C) A AC. ( C) A A CA;( C) A A CA. k ( A) ( ka) A( k). T ( A) T T A Mtries Lier Equtios Oe of the most importt motivtios for mtri multiplitio results from its lose oetio with systems of lier equtios. Cosier y system of lier equtios: m m m m Lier Alger I

5 Mtri Alger Mtri Multiplitio If A m m m, X, m These equtios eome the sigle mtri equtio: AX This is lle the mtri form of the system of equtios, is lle the ostt mtri. Mtri A is lle the oeffiiet mtri of the system of lier equtios, olum mtri X is lle solutio to the system if AX. The mtri form is useful for formultig results out solutios of system of lier equtios. Give system system: AX AX there is relte lle the ssoite homogeous system. If X is solutio to if X is solutio to AX, the X X is solutio to AX. Iee, AX AX, so: AX A( X X) AX AX This oservtio hs useful overse. Theorem Suppose X is prtiulr solutio to system The every solutio X to AX hs the form : X X X AX of lier equtios. for some solutio X of the ssoite homogeeous system AX. Proof: Lier Alger I 7

6 Mtri Alger Mtri Multiplitio Suppose tht X is y solutio to AX so tht AX. Write X = X - X, the X = X + X, we ompute: AX A( X X) AX AX Thus X is solutio to the ssoite homogeeous system AX. The importt of Theorem lies i the ft tht sometimes prtiulr solutio X is esily to fou, so the prolem of fiig ll solutios is reue solvig the ssoite homogeeous system. Emple Epress every solutio to the followig system s the sum of speifi solutio plus solutio to the ssoite homogeeous system. y z y z z Solutio Gussi elimitio gives t, y t, z t, where t is ritrry. Hee the geerl solutio is: t X y = t = + t z t Thus X = is speifi solutio, X = t gives ll solutios to the ssoite homogeeous system ( o the Gussi elimitio with ll the ostts zero) Theorem fouses ttetio o homogeeous systems. I tht se there is oveiet mtri form for the solutios tht will e eee lter. Lier Alger I

7 Mtri Alger Mtri Multiplitio Emple Solve the homogeeous system AX, where: A Solutio The reutio of the ugmete mtri to reue form is: So the solutio re r, t, r, t t y Gussi elimitio. Hee we write the geerl solutio X i the mtri form: X r t r t t = r + t = rx tx Where X X re prtiulr solutios etermie y the Gussi Algorithm. The solutios X X i Emple re lle the si solutios to the homogeeous system, solutio of the form rx tx is lle lier omitio of the si solutio X X. I the sme wy, the Gussi lgorithm proues si solutios to every homogeeous system AX ( there re o si solutio if there is oly the trivil solutio ). Moreover, every solutio is give y the lgorithm s lier omitio of these si solutios ( s i emple ) Lier Alger I 9

8 Mtri Alger Mtri Multiplitio Eerises.. Fi,,, if :.. 7. Verify tht A I A if:. A. A. Epress every solutio of the system s sum of speifi solutio plus solutio of the ssoite homogeous system.. y y y z z z.. Fi the si solutios write the geerl solutio s lier omitio of the si solutios.. e e e.. Let e mtri. Suppose A for some o zero m mtri A. Show tht o mtri C eists suh tht C I.. The tre of squre mtri A, eote tra, is the sum of the elemets o the mi igol of A. Show tht, if. tr ( A ) tra tr. tr ( ka) kta for y umer of k. tr ( T A ) tra. tr ( A) tr( A) A, re mtries: 7 7. A squre mtri is lle iempotet if P P. Show tht: e e e Lier Alger I

9 Mtri Alger Mtri Multiplitio., I re iempotets. If P is iempotet, so is I P P ( I P). If P is iempotet, so is T P. If P is iempotet, so is Q P AP PAP for y squre mtri A ( of the sme size s P ) 9. Let A e m e m. If A I the A is Iempotet.. Let A e igol mtries ( ll etries off the mi igol re zero ),show tht A is igol A= A Lier Alger I