Statistics for Financial Engineering Session 1: Linear Algebra Review March 18 th, 2006

Transcription

1 Sttistics for Ficil Egieerig Sessio : Lier Algebr Review rch 8 th, 6 Topics Itroductio to trices trix opertios Determits d Crmer s rule Eigevlues d Eigevectors Quiz The cotet of Sessio my be fmilir to my of you. However, it is very importt to uderstd bsic lier lgebr for future lectures.

2 . Lier Algebr Itroductio to trices.. trix: A mtrix is rectgulr rry of elemets rrged i rows d colums: A 3 3 Aother ottio for mtrices: A ij [ ] for i,; j,, 3 Note: Usully we cosider mtrices where the elemets of the mtrices re itegers or rel umbers. Note, however, tht we could cosider mtrices defied over y set. Tht set is clled the set of sclrs. (As we will see below, it is useful to hve opertios of dditio, subtrctio, d multiplictio defied o the set of sclrs tht mke up the elemets of the mtrices.) If mtrix A hs m rows d colums, we sy tht A is of dimesio m (or simply tht A is m mtrix). Exmple: 3 A is 3 mtrix. 3.. Squre mtrix: A mtrix is squre mtrix if its umber of rows is equl to its umber of colums, i.e., it is mtrix for some positive iteger. Exmple: 4 7 A is squre mtrix Colum d row vectors: A mtrix cotiig oly oe colum is clled colum vector, or simply vector. A mtrix cotiig oly oe row is clled row vector. I other words, colum vector is m mtrix, d row vector is mtrix, 4 7 Exmple: A is colum vector. 9

3 .4. Trspose: The trspose of mtrix A is other mtrix, deoted by A t, tht is obtied by iterchgig correspodig colums d rows of the mtrix A. Exmple: If 5 t 7 3 A 7, the A Note tht if A is of dimesio m, the A t is of dimesio m..5. Equlity of mtrices: Two mtrices A d B re sid to be equl if they hve the sme dimesio d ll their correspodig elemets re equl, i.e., ij b ij for ech i d j..6. Symmetric mtrix: If t A A, the A is sid to be symmetric. Exmple: 4 6 A 4 5 is symmetric Note tht symmetric mtrix must be squre mtrix..7. Digol mtrix: A digol mtrix is squre mtrix A whose offdigol elemets re ll zero, i.e., for i j. ij A 3.8. Idetity mtrix: The idetity mtrix (or uit mtrix) of dimesio is the digol mtrix whose elemets o the mi digol re ll (i.e., for ll < i ), d for i j. The idetity mtrix is deoted by I. ii ij 3

4 4 Exmple: The idetity mtrix of dimesio 4: I.9. Sclr trix: A sclr mtrix is digol mtrix whose mi digol elemets re the sme. Note tht sclr mtrix is sclr multiple of the idetity mtrix. Exmple: I.. Uit Vectors d mtrices: trices with ll uit elemets L O L J ' J ' L O L

5 . Lier Algebr trix Opertios.. trix Additio d Subtrctio Addig two mtrices, or subtrctig oe mtrix from other, requires tht they hve the sme dimesio. The mtrix dditio d subtrctio is defied s simply dditio d subtrctio of elemets i the correspodig positios of the two mtrices. A B A + B A B trix multiplictio: There re two types of mtrix multiplictio: ultiplictio by sclr λ : λ A Aλ [ λ ] ij ultiplictio of mtrix by other mtrix A 7 B AB Notes: trix multiplictio is ssocitive d distributive but ot commuttive, i.e., AB BA 5

6 .3. Defiitio (lier depedece): A set of c colum vectors C, C,C c is sid to be lierly depedet if oe of the vectors c be expressed s lier combitio of the others. If o vector of the set c be so expressed, we sy the vectors re lierly idepedet. A more geerl defiitio is: the c colum vectors re lierly depedet if c sclrs λ,λ,k,λ c, ot ll zero, c be foud such tht λ C + λ C +K+ λ c C c (where o the right hd side deotes the zero colum vector). Note tht c colum vectors C, C,C c form c mtrix, so we c pply the defiitios of lierly depedet d lierly idepedet to mtrices. Exmple: The mtrix 5 A 6 is lierly depedet, sice Rk of mtrix: The rk of mtrix is defied to be the mximum umber of lierly idepedet colums i the mtrix. Notes: The rk of r c mtrix cot exceed mi(r, c). If mtrix is product of two mtrices, its rk cot exceed the smller of the two rks of the mtrices beig multiplied. Thus, if C AB, the the rk of C cot exceed mib(rk A, rk B). We will revisit rk i the ext sectio. 6

7 .5. Iverse of mtrix: The iverse of mtrix A, deoted A -, is mtrix such tht A A - A - A I. Fidig iverse mtrices: A iverse mtrix of squre r r mtrix exists if the rk of the mtrix is r. Such mtrix is sid to be osigulr, or of full rk..5.. Iverse of mtrix of dimesio b d b A A c d d bc c.5.. Iverse of mtrix of dimesio 3 b c B d e f Let z (ek fh) b(dk fg) + c(dh eg) g h k The B A D G B E H C F K where A (ek fh)/z B ( bk ch)/z C (bf ce)/z D (dk fg)/z E (k cg)/z F (f cd)/z G (dh eg)/z H (h bg)/z K (c bd)/z For higher dimesios, we tur to Crmer s rule (ext sectio). 7

8 .6. Bsic theorems of mtrices: A + B B + A (AB)C A(BC) c(a + B) ca + cb λ(a + B) λa + λb ( A t ) t A ( A + B) t A t + B t ( AB) t B t A t ( ABC) t C t B t A t ( AB) B A ( ABC) C B A ( A ) A ( A t ) A ( ) t 8

9 3. Lier Algebr Determits d Crmer s rule 3.. Determit Clcultio 3... Determit of secod order: det( A ) 3... Determit of third order: det( A ) These s re clled miors. Cofctor of elemet is defied s of tht elemet. To expd determit i terms of its cofctors:. 3 det( A 3 ) C + C + 3C ) i+k ( times the mior Note tht the determit c be developed by y row or colum, i.e., it c be writte s the sum of y three elemets of row or colum ech multiplied by its cofctor Determit of rbitrry order: For rbitrry sized mtrix (x), the determit D is writte s: D, k ( ) i+ k ik Where i is,,.. or. ik Properties of Determits Some bsic properties: det(ab) det(a).det(b) det(a T ) det(a) 9

10 3.. Crmer s rule For system of equtios, give by: Ax b where A is mtrix, b d x re vectors. If det( A ), the the system hs precisely oe solutio, give s: Di xi D Where D i is determit obtied from D by replcig k th colum by the colum vector b Geerliztio of Crmer s rule - Iverse of mtrix Iverse of mtrix A A.. A A A.. A A det( A) A A.. A Where A jk is the cofctor of jk i det(a) Sigulr d No-sigulr mtrices For x squre mtrix A, the iverse A - exists if d oly of rk(a). Hece, A is o-sigulr if rk(a), d is sigulr if rk(a) <. oreover, if mtrix is o-sigulr, the iverse exists if d oly if: det( A )

11 4. Lier Algebr Eigevlues d Eigevectors 4.. Eigevlue problem Clcultio of Eigevlues/Eigevectors Cosider the vector equtio: Ax λx where λ is sclr umber. This c be writte s: ( A λ I) x Usig Crmer s theorem, this homogeous system hs otrivil solutio if d oly if det( A λ I) This is refereed to s the chrcteristic equtio or chrcteristic polyomil. The solutios of this equtios led to the differet eigevlues, i.e., the eigevlues re the roots of the chrcteristic equtio. Oce the eigevlues hve bee determied, correspodig eigevectors c be determied. Exmple: 5 A 4 The chrcteristic equtio c be writte s: 5 λ 4 D ( λ) det( ) λ λ λ The solutios re λ 6 d λ For the first eigevlue, the system tkes the form: x + 4x x x 4 The solutio is x 4x d therefore the first correspodig eigevector is: 4 x ) Similrly, the other eigevector is: x ) Note: The set of eigevlues is clled the Spectrum of A. The lrgest of the bsolute vlues of A is clled the Spectrl rdius of A.

12 5. Quiz ) Show tht det(ka) k det(a), where A is x mtrix d k is sclr. ) Let B be defied s follows: B Are the colum vectors of B lierly idepedet? b. Wht is the rk of B? c. Wht is the determit of B? 3) Let A. Clculte eigevlues ( λ,λ ) d eigevectors ( X, X ) b. Writig P [ X X ], Clculte B P AP. Wht is this mtrix B? c. Usig B d P, clculte A, where.