cosets Hb cosets Hb cosets Hc

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1 Exam, Do a total of 30 oits (moe if you wat, of couse) Show you wok! Q: 0 oits (4 ats, oits each fo the fist two ats, 3 oits each fo the secod two ats) Q: 0 oits ( ats, oit each, max cedit 0 oits) Q3: 0 oits (5 ats, oits each) Q4 : oits (6 ats, oits each) Q: Costucti a ou eesetatio based o cosets Let G be a fiite ou, ad H a subou of G Recall that the coset Hb is the set of all of the elemets of G that ca be witte i the fom h b, fo some elemet h i H, ad that the cosets costitute a atitio of G ito disoit subsets ow coside the fee vecto sace W o the cosets: that is, W be the vecto sace of fuctios fom the cosets Hb to Sice W is a fee vecto sace, it has a atual ie oduct, x, y å x( Hb) y( Hb), cosets Hb Fo ay fuctio x i W ad ay elemet of G, we defie Q, a membe of Hom( W, W ), as follows: Q takes x (a fuctio o the cosets) to the Q ( x ) (aothe fuctio o the cosets) whose value at the coset Hb is ive by ( ) Q ( ) ( ) ( ) x Hb x Hb A Show that this is a ou eesetatio B What is its dimesio (i tems of the sizes #( G ) of G, ad #( H ) of H)? C Suose futhe that G is commutative What is the chaacte of Q? D (o loe suosi that G is commutative) Ude what cicumstaces is Q ieducible? Solutios This set-u eealizes the eula eesetatio (it educes to the eula eesetatio whe H is the idetity subou), ad the aalysis oceeds alo the same lies A To see that Q is uitay: å ( ) ( ) å å Q (), x Q ( y) Q ()( x Hb) Q ( y)( Hb) x( Hb) y( Hb) x( Hc) y( Hc) x, y cosets Hb cosets Hb cosets Hc The cucial obsevatio is the thid equality: fo a fixed ou elemet, if Hb aes ove all cosets, samli each oe exactly oce, the so does Hb To see this, ote that if Hb ad Hb ovela, the some hb hb, fom which it follows that hb hb, ad that Hb ad Hb ovela also; sice the cosets ae disoit, if two ovela, they must be idetical Fomally, this is a chae of vaiables fom b to c b; b c - if c - samles each coset oce, the so does b To see that Q is a eesetatio ie, that QQ q Qq Q Qq( x) Qq( x) by evaluati the left ad iht had side at evey coset Hb We do this ust as i the aalysis of the eula eesetatio O the left, say y Q ( x) yhb ( ) Q( x) ( Hb) xhbq ( ) The ( ) Q ( ) ( ) ( ) ( ) y Hb y Hb x Hbq q, so ( q ) O the iht, ( ), we eed to show that ( ) Q ( ) ( ) ( ) q x Hb x Hbq diectly fom the defiitio of Q

2 B Sice the cosets divide a ou ito disoit subsets, thee ae #( G) / #( H ) cosets This is the dimesio of the fee vecto sace, sice the fee vecto sace cosists of all the fuctios o these cosets, ad theefoe is the dimesio of the eesetatio Fo C ad D: The chaacte is defied by c Q( ) t( Q) It is easiest to evaluate the tace usi the basis fo the fee vecto sace W cosisti of fuctios Hd which assi a value of to oe coset Hd, ad a value of 0 to othe cosets (This is also aallel to the situatio fo the eula eesetatio, whee we evaluate the tace i the fee vecto sace o G, athe tha i the fee vecto sace o the cosets) I this basis, Q acts as a emutatio matix To see this: Q Hd( Hb) Hd( Hb) This evaluates to if Hb ad Hd ae the same coset, ie, if Hb ad Hd ae the same coset So Q Hd Hd, ie, a elemet of the basis set Sice the tace is the sum of the diaoal elemet eties i the matix that descibes the tasfomatio, the chaacte c ( ) is the tally of the umbe of cosets Hd fo which Hd ad Hd ae idetical Q C Fo a commutative ou, the cosets Hd ad H d ae idetical (sice ad d commute) So if H, Q mas evey Hd to itself, sice Hd H d Hd (the last ste because H ) That is, fo H, Q, is the idetity tasfomatio,so fo H, c Q( ) t( Q) dim W #( G) / #( H) Covesely, if H, the H d ad Hd must be distict cosets, (sice if they ovelaed, the fo some h ad h, we would have h d hd which would imly h h ad that H ) So if H, Q mas evey Hd to a diffeet basis elemet, ad theefoe c Q( ) t( Q) 0 So i the case that G is commutative, c ( ) #( G Q ) / #( H ) fo H ad 0 othewise D As above, the chaacte is always a o-eative itee sice it is the tace of a emutatio matix The tace fomula fo the umbe of commo ieducible comoets of L ad M times is d( L, M) å cl( ) cm( ) Take L Q ad M E, the tivial eesetatio G c ( E ) eveywhee, ad ( Q ) 0 thouhout G, with ( e ) #( G Q ) / #( H ) 0 So dqe (, ) 0 So Q must cotai at least oe coy of the tivial eesetatio So the oly way that Q ca be ieducible is if it is, itself, the tivial eesetatio, which meas that it has dimesio, so #( H) #( G), so H G Questio Pemutatio matices ad eiedecomositios Coside a cyclic emutatio matix M, defied by m, fo ( ), m,, ad othewise zeo Hee, fo 5: M A Is M uitay? B Is M self-adoit? C What ae the eievalues of M? D What ae the eievectos of M?

3 E Coside a matix C, which is cyclic but with eties ot ecessaily daw fom {0,} (so it is ot a emutatio matix) Secifically, ck, fk, whee k is iteeted mod Fo examle, fo 5, f0 f f f3 f4 f4 f0 f f f3 C f3 f4 f0 f f Is C uitay? f f3 f4 f0 f f f f3 f4 f 0 F Is C self-adoit? G Does C commute with M? H What ae the eievectos of C? I What ae the eievalues of C? J What ae the eievalues of this matix, which is a emutatio matix but ot cyclic? P K What ca you say about the eievalues of ay emutatio matix of size? Solutios T * T A By diect calculatio, MM ( M M sice M is eal), so M is uitay B Excet fo, m, m, so M is ot self-adoit x x x x3 C ad D Say x is a eievecto The Mx, ad Mx x meas that x x, x x x x x3 x x,, x x x, ad fially x x x So, ad x is fee to vay That is, is a th oot of uity Fo ay such, ie, fo e E Tyically ot, as bei uitay * ( CC I) i would equie that * F Tyically ot, as C C would equie f f (amo othe this), the eievectos ae ive by x f (amo othe this) a

4 G Yes (by diect calculatio): MC CM ote that MC is a matix simila i stuctue to M but ow f is o the diaoal H Sice M ad C commute, the eievectos fo C ae the eievectos fo M I Choose a i the above exessio (at D) fo x i ad e Mx M ; the fist ety of the oduct ust be the fist eievalue; by diect multilicatio, this is i fe 0 M 3 0 J P is a block matix P That is, it acts seaately o a 3-dimesioal ad a 4-dimesioal 0 M 4 subsace, ad i each of these subsaces, it is a cyclic emutatio matix of the sot defied i at A O the 3-dimesioal subsace, its eievalues ae the same as those of M above with 3; o the 4-dimesioal subsace, its eievalues ae the same as of M above with 4 K They will always be th oots of uity fo some Questio 3 Tasfe fuctio ad owe sectum i a simle hysical system d x Say a outut, x( t ), is elated to a iut, st ( ) by m s-kx- c That is, the outut x is the ositio of a obect of mass m that is subect to a sum of thee foces: the iut sial s, a si-like estoi foce kx, ad a fictioal foce - c A Detemie the tasfe fuctio that elates x to s B Say s is white oise What is the owe sectum of x? C ad D Ude what cicumstaces does the owe sectum have a eak at a ozeo fequecy? At what fequecy is the eak? E Descibe what haes if c 0 (o fictio) Solutios A The elatioshi betwee x( t ) ad st ( ) is liea ad time-taslatio-ivaiat, so it eseves i t eiefuctios of the time-taslatio oeato, amely, the exoetials e w i t So if st ( ) e w, the x( t ) must be a multile of st ( ), ie, () ˆ x t L e iwt The ˆ( ) iwt d x iwl w e ad ( ) ˆ iwt ( ) ˆ iwt iw L w e - w L e d x Substituti fo each of the tems i m s-kx- c : ( ˆ iwt iwt ( ) ) ˆ iwt ( ) ˆ iwt m - w L w e e -kl w e - icwl e Solvi fo ˆ( L w ) yields ˆ( ) iwt iwt L w (- mw + k+ icw)( e ) e o, L ˆ( w ) - mw + k + icw

5 B I eeal, the iut ad outut owe secta of a liea system ae elated by P ( ) L ( ) P ( ) Taki P ( ) K, S PX ( ) K L( ) K m k ic PX ( ) K m k icm k ic This is K m kicm kic K K 4 m k c m c mk k C ad D The owe sectum has a eak at a ozeo fequecy whe the deomiato above has a miimum at a ozeo fequecy The deomiato is a quadatic i, ad i eeal, the quadatic Au Bu C B (with A 0 ) has a extemum at u Hee, A m ad B c mk, so thee is a miimum of the A mk c deomiato (ad a eak of the owe sectum) at This is ositive fo c mk, so if m c mk, thee is a eak i the owe sectum at a ositive fequecy (If this quatity is eative, the thee mk c is o eal 0 at which thee is a eak) The eak is at the fequecy m k E Whe c 0, thee is a fequecy at which L ( ) ad PX ( ) ae both ifiite ie, at which a m ifiitesimal iut will lead to a abitaily lae esose That is, thee is a esoace Questio 4 Coheece ad etwok idetificatio Say S () t ad S () t ae oise souces with owe secta P ( ) S ad which ae coected to two obsevable oututs R () t ad () filtes with tasfe fuctios L ( ) i X P ( ) S, ot ecessaily ideedet, R t by the followi etwok,whee L i ae liea S A Fid the owe secta P ( ) R ad P ( ) R of the oututs i tems of the owe secta PS i ( ) ad cosssectum PS, S( ) of the iuts B Fid the coss-sectum of the oututs C ad D Same as A ad B, but fo ideedet oise souces fully coected to obsevable oututs, athe tha two

6 E Defie S( ) as the coss-sectal matix of S, ie, the matix whose elemets ae ive by S PS ( ), i i Si ( ), ad similaly fo R Futhe defie L( ) PS, ( ), i S i as the matix of tasfe fuctios L ( ) i Wite a cocise exessio fo S( ) i tems of R( ) ad L ( ) F ow say we kow that each of the oise iuts ae ideedet, ad have a flat owe sectum, secifically, that S( ) I Ca we deduce the matix L ( ) fom the matix R( )? Why o why ot? Solutios A ad C (A is the same as C, but with ): Fist calculate Fouie estimates fo the esose Ri ( t ) : The, ( ) L ( ) s ( ) i ik k k æ öæ ö P lim lim L s L s çå è øèçå ø Ri T i i T ik k im m T T k m lim L L s s T ik im k m T k m L L lim s s ik im T k m k m T k m whee the last ste follows because the L s ae fixed values, uaffected by the limiti ocess So PR ( ) ( )lim ( ) ( ) i Lik w Lim w T sk w sm w k m T k m, whee we have used the otatioal covetio that L L P k m ik im Sk, Sm PS, ( ) ( ) k S w P m S w whe k m (this is cosistet because the coss-sectum of a sial with itself is its k sectum) B ad D æ öæ ö P lim lim L s L s çå è øè çå ø Ri, R T i T ik k m m T T k m, lim L L s s T ik m k m T k m L L lim s s ik m T k m k m T k m k m L L ik m P Sk Sm E Fom D, P ( ) ( * R, ( ) ( ) ( ) ( ) ( ) ( ) ) ( ) i R w Lik w Lmw PSk S w L L P m w w Sk S w, because the adoit is m the comlex couate of the tasose So k m k m ik m

7 * ( R ) ( L ) ( L ) ( S ), ad R L S ( L ) * i k m ik m km, ad ( ) * F Fom E, fo ideedet, white-oise iuts of uit owe, S( ) I R L L Oe caot deduce L ( ) fom R( ) The simlest easo is that eve fo (whee thee is o issue of chaels mixi), this caot be doe Hee, R L ( L ) * L The easo that L caot be detemied is that thee ae o-tivial filtes with U ( w ) : the ue delay, ad also, the examle of LSBB455b, homewok questio Relaci L by L ( w ) U ( w ), ie, covolvi L with U, does ot chae R L L U Fo, thee ae futhe easos that L caot be detemied, elated to the mixi of sials If U ( ) is ay uitay matix, the (by defiitio) U( ) U( ) * I, so ( ) ( ) ( ) ( ) * * * * That is, followi L by a etwok R L L L U U L L U L U that mixes the sials accodi to a uitay matix does ot chae the obseved coss-sectal matix R