symmetric/hermitian matrices, and similarity transformations

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1 Linar lgbra for Wirlss Communicaions Lcur: 6 Diffrnial quaions, Grschgorin's s circl horm, symmric/hrmiian marics, and similariy ransformaions Ov Edfors Dparmn of Elcrical and Informaion Tchnology Lund Univrsiy i Ov Edfors Diffrnial quaions ssum ha w hav a diffrnial quaion of h yp du u d u Givn an iniial valu u(0 ( h soluion bcoms u u 0 ( ( whr h marix xponnial is dfind d as ( I+ + +!! 0 If can b diagonalizd, w hav λ S Λ S S 0! 0! S λ Ov Edfors Diffrnial quaions (con. Diffrnial quaions sabiliy... and h soluion can b xprssd in rms of ordinary xponnials: λ u u S S u λ λ λ λ S c c x + + c x λ ( ( 0 ( 0 c S u ( 0 Linar combinaion of pur soluions dpnding only on ignvalus and corrsponding ignvcors From prvious slid: Λ ( λ u S S u c x + + c λ x 0 Sabiliy dpnds on h ignvalus: sabl if all ignvalus saisfy Rλ < 0 nurally sabl if som and all h ohr Sabl rgion R λ 0 R λ < 0 m Unsabl rgion R unsabl if a las on ignvalu has Rλ > Ov Edfors Ov Edfors 4

2 Marix xponnial a warning Marix xponnial how o calcula Many propris of h marix xponnial ar analogus o h propris of ordinary xponnials... bu hr ar diffrncs! Exampl: Ordinary xponnial Marix xponnial a+ b a b + B B Equaliy holds, howvr, if and B commu, i.., if B B. Thr ar many suggsd mhods for calculaing marix xponnials. Diagonalizaion is on way, if h marix in h xponn can b diagonalizd: S Λ S SΛS If is nil-pon, i.., m 0 for som m, w can us h dfiniion (fini sum: m 0! Ov Edfors Ov Edfors 6 Marix xponnial how o calcula Marix xponnial - drminan If is a projcion marix, i.., if h dfiniion givs: I +!!! 0 0 I+ I+! ( For h full ramn: C. Molr & C van Loan. inn Dubious Ways o Compu h Exponnial of a Marix, Twny-Fiv Yars Lar. SIM REVIEW (lcronic publ Ov Edfors 7 Th drminan of an marix xponnial is: If can b diagonalizd: d r( ( poran consqunc: marix xponnial is always invribl, sinc is drminan canno b zro! Λ Λ Λ ( ( S S ( S ( ( S ( d d d d d d λ d λ ( λ λ λ ( λ λ λ r Ov Edfors 8

3 EIGEVLUES RE IMPORTT! Grshgorin s circl horm W hav sn ha ignvalus ar imporan in many siuaions. Thy drmin h sabiliy of boh diffrnc and diffrnial quaions... hy can hlp us o find soluions o many yproblms formulad in rms of marix quaions... c. To g som mor fling for h ignvalus of marics, l s a a loo a Grshgorin s Cricl Thorm. [Only brifly xplaind in h xboo.] Givn an x marix dfin h following sums of magniuds of off-diagonal lmns in ach row: R a, l l ow, ach ignvalu of is in a las on of h following (Grshgorin diss: l { z : z a, R } This horm can b usd o quicly bound h magniud of ignvalus, in som siuaions find ou if a marix is non-singular (if all ignvalus ar non-zro, if a sysm of diffrnc or diffrnial quaions ar sabl (no always possibl, c Ov Edfors Ov Edfors 0 Grshgorin s circl horm (con. Th Grsgorin circl horm is suprisingly simpl o prov. L x b an ignvcor of an x marix [a m,n ], wih corrsponding ignvalu λ, i.., x λx. Furhr, assum ha h h lmn x of x is h on wih h largs magniud (canno b zro!. x λx now givs (loo a lmn : al, xl λx al, xl λx a, x ( λa, x l l l xl λ a, x al, xl λ a, al, l l x l l λ a, al, DOE! l l Ov Edfors Grshgorin s circl horm (con. Us h horm on h following marix: Conclusions: - is invribl (by rows - o ignvalu has magniud largr han 7 (by columns. - o ignvalu has a magniud lss han (by rows.... mor? diagonal lmns of ignvalu of By rows R a l l, l R R R By columns Rl a l, l R R R Ov Edfors R R

4 Grshgorin s circl horm (con. Grshgorin s circl horm (con. W now ha all ignvaus mus b in a las on of h Grshgorin diss, i.., all ignvalus ar found SOMEWERE ISIDE TE UIO of h Grsgorin diss. Can w say anyhing mor abou whr ignvalus ar? Exampl (fully joind union: Exampl (hr disjoin ss n xra rsul concrning h locaion of ignvalus in h Grshgorin diss: If h union of all Grshgorin diss consiss of svral disjoin subss, ach such subs conains a numbr of ignvalus corrsponding o h numbr of diss forming h subs. Exampl on prvious slid: Thr ignvalus in his subs! R R R Th ignvalus can b anywhr in h subs! Whr ar h ignvalus in hs cass? On ignvalu in ach of hs! Ov Edfors Ov Edfors 4 Grshgorin s circl horm (con. Grshgorin s circl horm (con. simpl illusraion i of how o prov h rsul on h prvious slid: n arbirary squar marix can b wrin in h form: a a a 0 a + a a a a 0 ow, dfin a marix G(α as D diagonal nris ( α + G D α B whr α is a ral numbr in h inrval [0,]. W hav: G ( 0 D G ( B off-diagonal nris Ov Edfors 5 W now hr hings:. Th ignvalus of G(0 ar xacly h diagonal nris (of.. Th ignvalus of any marix ar coninuous funcions of h marix nris (or α in his cas. If marix lmns chang smoohly, hr ar no jumps in h ignvalus. 3. For ach valu on α, Grshgorins circl horm lls us in which s all ignvalus of G(α ar. ow, h ric: smooh ransiion from α 0oα α R R R α 0 α 0.5 α Th ignvalus ha sar a h diagonal nris (α 0 canno scap o anohr disjoin subs of h Grshgorin diss, sinc hy ar coninuous funcions of α. DOE! Ov Edfors 6

5 rmiian marics rmiian marix i has h propry Symmric/hrmiian marics rmiian (or ral symmric marics hav h following propris:. x x is ral for all complx vcors x.. vry ignvalu is ral. 3. wo ignvcors coming from diffrn ignvalus ar orhogonal Ov Edfors Ov Edfors 8 rmiian marics [con.] rmiian marics [con.]. x x is ral for all complx vcors x.. vry ignvalu is ral. L s s wha w g whn w a h complx conjuga of h scalar x x: ( x x x x x x x x x x quals is own complx conjuga > REL! W hav x λ x muliplying by x from h lf givs x x λ x x Ral Ral Prop. which forcs λ o b ral Ov Edfors Ov Edfors 0

6 rmiian marics [con.] Uniary marics 3. wo ignvcors coming from diffrn ignvalus ar orhogonal. W hav x λ x and x λ x W can show ha ( λ ( x x x x x x λ x x x λ x or (ignvalus ar ral, Prop. λx x λxx which for diffrn ignvalus can only b ru of x and x ar ohogonal Ov Edfors Uniary marics ar h complx quivalns of ral orhogonal marics. ral orhogonal marix Q has orhonormal columns: T QQ I complx uniary marix U has orhonormal columns: U U I Ov Edfors Spcral horm Spcral horm [con.] Th prvious propris for complx rmiian marics lad o on of h mos imporan rsuls in linar algbra h spcral horm: REL CSE ral symmric marix can b facord ino QΛQ T. Th oronormal ignvcors of ar in h orhogonal marix Q and h corrsponding ignvalus in h diagonal marix Λ. Th spcral horm implis ha a rmiian marix λ u u u λ n u COMPLEX CSE rmiian marix can b facord ino UΛU. Th oronormal ignvcors of ar in h uniary marix U and h corrsponding ignvalus in h diagonal marix Λ. Th abov ar simply spcial cass of h gnral rsuls on marix diagonalizaion whn ignvalus ar disinc. Thy can, howvr, b provn ru also for rpad ignvalus. can b wrin as a sum of ran- projcion marics: λ n u n u n n If h ignvalus of (nonngaiv, ral ar sord in dcrasing ordr, w can ma good low-ran approximaions of by limiing h numbr of rms w us in h abov sum Ov Edfors Ov Edfors 4

7 Similariy ransformaions W hav discussd svral forms of facoring marics basd on ignvalus and ignvcors: Similariy ransformaions SΛS T QΛQ QΛQ UΛU UΛU Λ S S [Diagonalizaion] Λ Λ Q Q [ symric] [ rmiian] U U Ths ar all ar all on h form: M M which w call a similariy ransformaion of. W say ha M - M is similar o, bu in wha way? Ov Edfors Ov Edfors 6 Similariy ransformaions [con.] If B M - M, hn and B hav h sam ignvalus and vry ignvcor x of corrsponds o an ignvcor M - x of B. ssum x λx and MBM - hn B has sam ignvalu as... x MBM x λx BM x λm x Som nos on applicaions of rmiian marics... bu a nw ignvcor Thr is mor in h xboo b abou his if you ar inrsd! Ov Edfors Ov Edfors 8

8 Corrlaion ssum ha x is a zro-man sochasic vcor. Th auocorrlaion is dfind as h xpcaion: Rxx E { xx } This auocorrlaion marix is rmiian and can hrfor b facorizd as R xx UΛU Givn his spcral facorizaion w can ma h following ransformaion This is calld h olling y U x ransform of x. (c.f. h Karhunn- Low ransform which rsuls in a nw, zro-man, sochasic vcor y wih auocorrlaion marix { } { } { } R E yy E U xx U U E xx U yy xx U R U U UΛU U Λ Elmns of y ar uncorrlad! Ov Edfors 9 Sysm modl x 0, x, x, T -poin IDF s 0, s, symbol m sampl n subcarrir s( L CP lngh T samp sampling priod ralll o s rial CP h TX - s, Par 3 -poin IDFT:, mn sm xn, xp jπ for 0 m n 0 dding CP: s s + for L m TX filring: m, m, ( ( h * TX s m, δ ( + m L s ( L T samp mt samp h TX 3 TX filr L Ov Edfors 30 Sysm modl [con.] s( Channl h ch ( } T ch ois n s ( r ( ( r( s( * hch ( + n( CP CP CP CP } LT samp T ch s long as h CP is longr han h dlay sprad of h channl, LT samp > T ch, i will absorb h ISI. By rmoving h CP in h rcivr, h ransmission bcoms ISI fr Ov Edfors 3 Sysm modl [con.] r ( RX filring: h RX T samp CP 3 ( h ( r( z RX * Sampling: z z ( T samp rial o para alll Sr r 0, y 0, q symbol r, p sampl y, n subcarrir r, -poin DFT Rmoving CP: rp, q z q( + L + p for 0 p -poin DFT:, np yn q rp, q xp jπ for 0 n p 0 y, L CP lngh T samp sampling priod RX filr h RX 3 L Ov Edfors 3

9 Sysm modl [con.] Sysm modl [con.] 0, n 0, x 0, y 0, x, Simplifid modl undr idal condiions (slow nough fading and sufficin CP Toal filr in h signal pah: * *, n hsignal, ( htx ( h( hrx ( ( f ( f * ( f * ( f y, signal TX RX W hav ndd d up wih an marix modl: y Xh + n whr y is h rcivd vcor, X a diagonal marix wih h ransmid consllaion poins on is diagonal, h a vcor of channl anuaions, and a vcor n of rcivr nois. For h purpos of channl simaion, assum ha all ons ar ransmid, i.., ha X I. W now hav a simplifid modl: sysms hav bwn 64 (WL and 89 (DigTV. Givn ha subcarrir n is ransmid a frquncy f n h anuaions bcom: n, signal ( f n y h+ n Furhr assum ha h channl is zro-man and has auocorrlaion R hh whil h nois is i.i.d zro man complx Gaussian wih auocorrlaion R nn σ I. W also assum ha h and n ar indpndn Ov Edfors Ov Edfors 34 Channl simaion Channl simaion [con.] Whn w rciv y, w wan o apply a linar asimaor h y ha minimizs h man-squard rror MSE E E{ h h } This givs h minimizing marix ( σ R R R R R R + I hy yy hh yy hh hh Evn wih nown R and a rsuling linar MMSE simaor hh and σ, maing a prcalculad marix possibl, ach channl simaion rquirs + σ hh ( hh h R R I y hh hh y opraions Ov Edfors 35 Can w simplify h calculaion? L s us h fac ha R hh is rmiian and can b facord as UΛU : ( σ ( σ ( ( + σ ( σ y ( σ h UΛU UΛU + I y UΛU UΛU + UU y UΛU U Λ + I U y UΛU U Λ + I U y UΛ Λ+ I U y Diagonal marix In raliy, i urns ou ha U is vry clos o h FFT marix Ov Edfors 36

10 Channl simaion [con.] Channl simaion summary Wha hav w obaind? hh ( hh +σ R R I U ( +σ I Λ Λ U 0 8 y Full x h y - Diagonal x - h marix poin marix poin muliplicaion. IFFT muliplicaion. FFT COMPLEXITY (opraions log + + log + log Esimaion complxiy (opraion ns umbr of subcarrirs [] Complxiy gain can b svral ordrs of magniud for larg sysms Ov Edfors Ov Edfors 38

Midterm exam 2, April 7, 2009 (solutions)

Midterm exam 2, April 7, 2009 (solutions) Univrsiy of Pnnsylvania Dparmn of Mahmaics Mah 26 Honors Calculus II Spring Smsr 29 Prof Grassi, TA Ashr Aul Midrm xam 2, April 7, 29 (soluions) 1 Wri a basis for h spac of pairs (u, v) of smooh funcions