Matrix Theory, Math6304 Lecture Notes from October 23, 2012 taken by Satish Pandey

Transcription

1 Matrix Theory, Math6304 Lecture Notes from October 3, 01 take by Satish Padey Warm up from last time Example of low rak perturbatio; re-examied We had this operator S = M (C) S is said to be the Cyclic shift operator ithesesethattheactioofs o ay vector x =(x 1,x,x 3,,x ) T C yields Sx =(x,x 3,,x,x 1 ) T which is obtaied by movig the first etry of x to the ed, (i.e. Sx is x cycled forward oe step). Observe that 0 1 S 1... = M (C) Cosequetly, x C, S x is precisely x cycled backward oe step. It follows the that S S = SS = I thereby rederig S Uitary. ItisalsoevidetfromthecylicpropertyofS that S = I (as S x is precisely x cycled forward step ad for ay x C,thisactiopreservesits terms ad hece reproduces x) We the proceed to uravel the eigevalues of S. RecallthattheeigevaluesofS are precisely the roots of the characteristic polyomial equatio We have the λi S = det(λi S) =0 λ λ

2 Expadig out alog the first colum of the above matrix we obtai det(λi S) =λ(λ 1 )+( 1) +1 ( 1)( 1) 1 = λ +( 1) +1 = λ 1 Thus the eigevalues of S are precisely λ j = e πij/,j {0, 1,,, 1} The eigevalues of S are of much iterest as they provide a aid to visualise the eigevalues of S. Let λ be ay eigevalue of S the for some ozero x C we have Sx = λx = S Sx = λs x = x = λs x = S x = 1 λ x = λx where the secod equality i the last implicatio is a direct cosequece of the fact that λ is uimodular. But λ = e πij/ which agai happes to be oe of the th roots of the uity. Sice there are distict eigevalues of S ad the λ with which we bega our argumet is arbitrary, it follows that S also has distict eigevalues. But the the spectrum of S turs out to be the same as the spectrum of S. 1 Kowig the eigevalues of S it is ot hard to fid the spectrum of H := S + S.Notethat S ad S have the same eigevectors so if x C is a eigevector of S the Hx =(S + S )(x) =S(x)+S (x) = λ(x)+λx =(λ + λ)x =Re(λ)x where λ {e πij/,j {0, 1,, 1}} This shows that x is also a eigevector of H ad the correspodig eigevalue is λ j := e πij/ + e πij/ πj =cos 1 It is imp to ote that this is ot true i geeral. that spectrum of T = spectrum of T

3 for some j {0, 1,, 1}. I fact, the eigevalues of H are precisely λ j. (without orderig i accordace with size) To see this we just eed to covice ourself that S has distict eigevalues ad thus it s eige vectors (ad hece the eige vectors of H) areliearlyidepedet. ThusH caot have ay other eige value, for if it does the the correspodig eige vector will joi the set of already existig liearly idepedet eige vectors of H thereby givig a liearly idepedet set of vectors with cardiality +1,whichisotallowed. Sketch of eigevalues of H; Orderig accomplished I what follows, we will, via graphical tools, observe the eigevalues of H idexed by j ad make a attemp to liearly order them (coutig multiplicities). I the precedig sectio, we observed that the eigevalues of H are precisely πj λ j =cos,j {0, 1,, 1}. Needless to say the period of the above fuctio is ad hece it suffices to study its graph o the iterval [0,) j j Let us ow try to abstract further iformatio from the above graph. For each j {0,, 1} the poit (j, λ j ) o thegraph correspods to a eigevalue of H. Sice the Cosie curve i [0,) is symmetrical about the vertical lie x =,itfollowsthe that λ j = λ j wheever 0 <j< ; j N; adthusallsuch λ j s have multiplicity. The eigevalues λ 0 ad λ / (appears as a eige value oly whe is eve) are both of multiplicity 1. λ 0 =ad if is eve the λ / =. We are ow ready to order the eigevalues of H. To accomplish this we distiguish cases: Observe that H =(S + S )=S + S = S + S = H. SiceH is Hermitia its eigevalues are real ad thus we ca order them. 3

4 case-i: whe is eve. Clearly the is a iteger ad hece λ / = is the smallest eige value of H. Followig the covetioal (deservedly popular) otatio, λ k (H) = kth smallest eige value of H, wesay λ 1 (H) =. Further it is easy to covice ourselves that λ k (H) =λ k+1 (H); fork eve ad k. Observethat λ k (H) = λ / k/ =cos(π( k )/) =cos(π kπ ) = cos( kπ ). Ad lastly, λ (H) = λ 0 =. case-ii: whe is odd. We the have / o-itegral ad hece the smallest eige value of H doubles givig Also, Ad fially, λ 1 (H) =λ (H) = λ / 1/ =cos(π( 1 )/) = cos(π/) λ k (H) =λ k+1 (H); k odd,k λ (H) = λ 0 =. = λ k =cos(π k /). We coclude this piece of discussio by a ote that the above formulae are ice whe k ad are both eve or odd. We ext cosider the followig matrix H := ad study the relatio betwee the eigevalues of H ad H. (Seethefootote) 3 ; 3 The matrix H whe cosidered as a operator o C has a special ame Dirichlet Laplacia operator. 4

5 Note that H := H where the matrix i the secod term is a rak matrix with eigevalues {0, ±1}.Clearly the H is a rak- perturbatio of H. As we will soo see, beig a low-rak perturbatio, the eigevalues of H will ot differ by much from the eigevalues of H. Bythetheoremorakr perturbatio (Thm.4.3.8) with r =,wehave or, λ k (H) λ k+ ( H) λ k+4 (H); k 4 λ k (H) λ k ( H) λ k+ (H); k It readily follows from above that if ad k are both odd or both eve ad k, the we have π(k ) cos( ) λ k ( H) π(k +) cos( ) This iequality illustrates that for large values of wehavecotroloalmostalloftheeigevalues of H, whichisexactlywhatweexpected.nowif is odd ad k is eve or is eve ad k is odd ad 3 k 1, weobtai cos( π(k 3) ) λ k ( H) π(k +1) cos( ). (Note that because of the eigevalue doublig, the argumet of the term o the extreme left of the above iequatio assumes π(k 3)/ rather tha π(k )/) The iequalities give above supports the expectatios with which we bega the discussio ad ifers that most of the eigevalues of H udergo a very little chage. 4.5 Eigevalues of matrices ad pricipal submatrices Last time we geeralized the iterlacig theorem ad exteded our discussio o the kid of eigevalues that ca be created by puttig two Hermitia operators with differet eigevalues. The study of the relatioship betwee eigevalues of matrices ad eigevalues of their sum is, ideed, a exceptioally iterestig area to go through the mid s eye, ad it becomes eve more useful whe the matrices i questio are ot compatible for additio (i.e. they are of differet orders). This motivatio results i what follows. As a applicatio of the Courat-Fisher theorem, we shall ow retur to Weyl s isight ad relate the eigevalues of a Hermitia matrix ad it s pricipal submatrix. 4 We will begi with the 4 A submatrix obtaied by deletig several rows ad the correspodig colums from a give Hermitia matrix A is said to be a pricipal submatrix of A. Note that ay pricipal submatrix of a Hermitia matrix is Hermitia. 5

6 simplest case whe they are related by borderig.(i.e. addig a ew last row ad colum) Cosdier a Hermitia matrix A M (C) ad border it by addig a ew last row ad colum to it such that the ewly geerated matrix, say Â, preserveshermiticity. A. y  = M +1 (C) y. a It s ot hard to see that for  to be Hermitia we essetially require a tobeit sowtraspose cojugate ad hece real ad y to be the cojugate traspose of y. Wethehavethefollowig Iterlacig Eigevalues Theorem for bordered matrices. Theorem: Let A M,  M +1 be Hermitia ad related by borderig, a =[Â] +1,+1. Deote the eigevalues of A by {λ j } j=1 ad of  by {ˆλ j } +1 j=1 arraged i o-decreasig order λ 1 λ... λ ad ˆλ 1 ˆλ... ˆλ +1.The [This Thm is a aalog of rak-1 perturbatio] ˆλ 1 λ 1 ˆλ λ... λ ˆλ +1. Proof. Sice we have to relate the properties of A ad Â, itimplicitlyidicatesthatwehaveto somehow jump betwee the spaces they sit i. O this pretext let S k ad Ŝk deote k-dimesioal subspaces of C ad C +1 respectively. Usig the strategy of Courat - Fisher, we deduce ˆλ k+1 =mi Ŝ k+1 mi Ŝ k+1 mi S k = λ k 0=ˆx C +1 ˆx S k+1 0=ˆx C +1 ˆx S k+1 ˆx e +1 ˆx, ˆx ˆx ˆx, ˆx ˆx Ax, x 0=x S k x where the first iequality results from the additioal restrictio ˆx e +1 imposed o ˆx [See footote] 5 ad the secod iequality is the cosequece of the restrictio of ˆx S k+1 to the k-dimesioal subspace S k.[see footote] 6 5 Observe that if ˆx S k+1 is already orthogoal to e +1 the the dimesio of S k+1 is k +1 ad if ot the this orthogoal restrictio shriks the dimesio of S k+1 to k, thereby reducig the imum. Also, this orthogoality restrictio o ˆx eables us to replace  by A. 6 Sice the imum over a bigger set decreases, restrictio of ˆx to S k further lowers the imum. Also, this impositio of the restrictio replaces ˆx by x. 6

7 Similarly, ˆλ k = mi Ŝ k+1 0=ˆx Ŝ k+1 Ŝ k+1 S k = λ k mi 0=ˆx S k+1 ˆx e +1 ˆx, ˆx ˆx ˆx, ˆx ˆx Ax, x mi 0=x S k x where the first iequality, as i previous case, results from the additioal restrictio ˆx e +1 imposed o ˆx for it implies that the dimesio of Ŝ k +1is either k =1or k ad hece the miimum goes up. For the secod iequality, the obvious reaso is the restrictio of ˆx Ŝ k+1 to a lower dimesioal subspace Ŝ k, forchoosigalowerdimesiosubspaceŝ k forces the set to be smaller over which miimum is supposed to be evaluated thereby further icreasig the miimum. This eds the proof. The precedig theorem is the first hit of a geeral situatio, as of ow it deals with the situatio i which a Hermitia matrix is bordered by addig a ew last row ad colum, but they could also be thought of as givig iformatio about the behaviour of the eigevalues of ahermitiamatrixwheit slastrowadcolumare deleted. There is, of course, othig special about last row ad colum. If the ith row ad colum of the matrix  i above theorem are deleted istead of the ( +1)rst, oe merely chages e +1 to e i the proof ad obtais the same iterlacig iequalities.this is exactly what the ext corollary states. Corollary: If A M is a pricipal submatrix of  M +1,  =  ad {λ j } j=1, {ˆλ j } +1 j=1 are respectively their eigevalues, the ˆλ j λ j ˆλ j+1 for j {1,, 3,...,}. Proof. If A,  are related by borderig, we are doe by the previous theorem. Otherwise, permute rows ad colums accordigly, which does ot chage eigevalues. The rest is trivial from the discussio made i the paragraph precedig the corollary. To have a quatative isight of the precedig corollary, we shall go through a example. Let  M +1, =  ad  = Â. Also assume that rak (Â) =r. Itshouldotbetoughat this stage to see that the eigevalues of  are 0 ad 1. Sice is Hermitia,  is diagoalizable by a uitary chage of basis. Thus there exists U such that UÂU = D with D diagoal. It 7

8 follows the r =rak(a) =rak(d) =Totalumberofozeroetriesothediagoalofthe matrix D =Totalumberof1 appearig o the diagoal of D. Hecemultiplicityof1 is r ad that of 0 is +1 r. We ow choose a pricipal submatrix A M ad wish to abstract as much iformatio as possible related to the eigevalues {λ j } j=1. Thebytheprecedigcorollary,wehave 0 λ 1 0 λ 0... λ r 0 λ r+1 1 λ r λ 1 which implies that λ 1 = λ =... = λ r = 0 ; λ r+ = λ r+3 =... = λ = 1 ad 0 λ r+1 1. Hece we happe to kow precisely all but oe eigevalues of A ad also figured the boud o the remaiig oe. Needless to say, this example explicitly delieates the stregth of the precedig Theorem/Corollary. Its high time sice we dealt with the relatio of the eige values of a Hermitia matrix with oe of its pricipal submatrices with dimesio oe less tha that of the origial matrix. What ca be said about this relatio whe the pricipal submatrix i questio has a differece i dimesio, of more tha 1, whecomparedtotheorigialmatrix? Thisaggigquestiomotivates the followig theorem which is i some sese the aalog of rak r-perturbatio: Theorem: Let A M, A = A. Also assume r N, r ad A r M r be a pricipal submatrix of A. The for each k N ad k r we have Proof. To be doe i the ext class. λ k (A) λ k (A r ) λ k+ r (A). 8