THE SPECTRAL RADII AND NORMS OF LARGE DIMENSIONAL NON-CENTRAL RANDOM MATRICES

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1 COMMUN. STATIST.-STOCHASTIC MODELS, 0(3), (994) THE SPECTRAL RADII AND NORMS OF LARGE DIMENSIONAL NON-CENTRAL RANDOM MATRICES Jack W. Silverstei Departmet of Mathematics, Box 8205 North Carolia State Uiversity Raleigh, North Carolia Key words ad phrases: Radom matrix, spectral orm, spectral radius. ABSTRACT Cosider a matrix made up of i.i.d. radom variables with positive mea ad fiite fourth momet. Results are give o its spectral orm ad (if it is square) spectral radius as the dimesio icreases.. INTRODUCTION I your favorite computer laguage, create a matrix U full of i.i.d. radom variables, uiformly distributed o (0, ). Compute, the Perro eigevalue, the real eigevalue of U equal to the spectral radius (the maximum, i absolute value, of its eigevalues), guarateed to exist for positive matrices. It turs out to be ear 50. Why? The aswer depeds o the followig result.

2 526 SILVERSTEIN Theorem. ([],[4]). For =,2,...,ads=s()forwhich/s y>0as, let V =(v ij ), i =,2,...,, j =,2,...,s,where v ij, ij =,2,...,, are i.i.d. radom variables with E(v )=0ad E(v 2 )=σ2 <. The the spectral orm s V (where for ay rectagular matrix A, A equals the square root of the largest eigevalue of AA T ) coverges a.s. to ( + y)σ as E(v 4 ) <. If E(v)=, 4 the lim sup s V = a.s. The matrix U is of the form V +µe e T,whereV is as above with s =, µ>0, ad e = (,,...,) T. View U as a perturbatio of µe e T, a rak-oe matrix with positive eigevalue µ, ad exploit a perturbatio theorem, such as Corollary i [3]: ˆ eigevalue of A + E, A ormal = the existece of eigevalue i of A such that ˆ i E. Thus, whe µ is evetually greater tha 2 V (which occurs a.s., sice V 2σ a.s.), a simple cotiuity argumet applied to the eigevalues of tv + µe e T, t [0, ], will yield, the largest (i absolute value) eigevalue of V + µe e T, to be real, positive, with multiplicity, ad µ K with K a.s. 2σ as. (.) (Follow the cotiuously chagig eigevalues of tv + µe e T as t moves from 0 to. For fixed t they all must lie i the uio of the two disjoit discs i the complex plae cetered at the origi ad µ, both havig radius t V. Necessarily, for all t [0, ], oe ad oly oe eigevalue of tv +µe e T ca lie i the disc cetered at µ with radius V, ad it must remai real, positive, ad larger tha all the other eigevalues i absolute value.) We see ow where 50 comes ito play, sice µ for our U is simply /2. But seems to be much closer to 50 tha what is quarateed by (.). Ideed, a simulatio of 000 geeratios of idepedet U s resulted i Perro eigevalues ragig betwee ad 5.3. (.) would place merely betwee ad (usig the fact that the variace of a uiformly distributed r.v. o (0, ) is /2). Compute the spectral orm of U. It caot be smaller tha. A relatio correspodig to (.) ca be derived usig similar cotiuity

3 SPECTRAL RADII OF RANDOM MATRICES 527 argumets from a perturbatio theorem o the sigular values of rectagular matrices, such as Corollary i [3]: Let A ad B be s rectagular matrices, with respective sigular values σ σ 2 σ q,adτ τ 2 τ q,whereq=mi(, s). The for all i =,2,...,q, It follows that σ i τ i B A. µ s K with K a.s. ( + / y)σ as, (.2) where = V +µ se e T s, adv, s,yare defied as i Theorem.. But, agai, simulatios show U to be much closer tha 0/ 3 away from 50. The purpose of this paper is to provide more detailed iformatio o the limitig behavior of the spectral radii ad spectral orms of radom matrices as the dimesio icreases, with etries havig positive meas. The followig theorems will be proved. Theorem.2. Let be the largest (i absolute value) eigevalue of V + µe e T,whereµ>0, ad V is defied i Theorem. with s = ad E(v 4 ) <. The, with probability oe, is real ad positive (that is, it is the spectral radius of V + µe e T ) for all sufficietly large. Moreover, = µ + X + Z where {Z } is a tight sequece, ad X = cetral limit theorem, µ D N (0,σ 2 ). i,j v ij. Thus, by the Theorem.3. WithV s,ydefied as i Theorem., E(v)< 4 ad µ>0 V +µ se e T s = µ s + ( σ 2 ) s 2 µ s + + X + Z where {Z } is tight, ad

4 528 SILVERSTEIN X = s i j s v ij D N (0,σ 2 ). These theorems obviously accout for the small variatio of the Perro eigevalue ad spectral orm of U about 50, the latter beig approximately σ2 µ =/6 larger tha the former. Notice results for egative µ ca be trivially derived from these theorems. The proofs, give i the ext sectio, rely maily o Theorem., (.), ad (.2), ad require little additioal probabilistic argumets. They ca easily be exteded to allow for complex etries i V (the proof of Theorem. ca be modified to the complex case). As will be see, Z i either case ca be expressed i a form for which further aalysis is possible. A more detailed study of Z will udoubtably yield it to be asymptotically ormal. It is remarked here that a similar result is obtaied for the largest eigevalue of o-cetral radom matrices of Wiger type, that is, symmetric matrices with idepedet etries o ad above the diagoal ([2]), although with a proof more probabilistic i ature. The techiques used i the ext sectio ca easily be applied to the Wiger case. 2. PROOFS OF THEOREMS.2,.3 For the followig, Z will deote a geeric radom variable, ot ecessarily the same quatity from oe appearace to the ext, for which {Z } is tight. We start with V, satisfyig the coditios of Theorem.2. Cocetratig for the momet o realizatios for which V 2σ, for ay fixed realizatio we assume is large eough so that, defied i Theorem.2, satisfies V /2. Notice, the, I V is ivertible, ad (I V ) 2(use the fact that, for square A, (I A) = j=0 Aj wheever A < ). Let f be a eigevector of V + µe e T correspodig to.thev f+e T fµe = f,which implies µe T f( I V ) e = f. Multiplyig o both sides by e T,we fid (otig that e T f caot be 0) = µe T (I V ) e. Write

5 SPECTRAL RADII OF RANDOM MATRICES 529 = µ( + e T V e + e T 2 V 2 e + e T ( V ) 3 (I V ) e ). We have µe T ( V ) 3 (I V ) e 2µ ( V ) 3 = K, where K 6( σ µ )3. For all ad all realizatios, let { µ 3/2 e T Y = ( V ) 3 (I V ) e if V /2 ( µ( + e T V e + e T 2 V 2e )) o.w. The, sice Y sup k tightess of {Y }. We have Y k lim sup Y 6( σ µ )3 a.s., we have the E(( e T V 2 e ) 2 )= 3E(( ijkv ij v jk ) 2 )= 3(E(v4 )+(3( ) ( ))σ 4 ). Therefore, { e T V 2 e } is tight, which implies { µ3/2 e T 2 V 2 e + Y } is tight. At this stage we have = µ + µ X + Z (X defied i Theorem.2). Therefore, µ = Z,adtheproofofTheorem.2 is ow complete. We proceed to the proof of Theorem.3, where V is s. To facilitate the expositio, we will write V = V.Let= V+µ se e T s. Write V + µ se e T s i its sigular value decompositio UΛV ([3], p. 45), where U is, V is s s, both orthogoal, ad Λ is o-egative diagoal, its diagoal elemets arraged i o-icreasig order. The =Λ.Letu,vbe the first colums of U,V T, respectively. The Vv+e T svµ se = u ad V T u + e T uµ se s = v. For the momet we cocetrate o a realizatio for which V ( + y)σ s,adsufficietly large so that V 2.The either e T u or et s v will be 0, 2 I VV T, 2 I V T V are ivert-

6 530 SILVERSTEIN ible (I deotig the geeric square idetity matrix), ad max( (I 2 VV T ), (I 2 V T V) ) 2. We have V T Vv+e T svµ sv T e = v e T uµ se s. Therefore which implies µ s(e T s vv T e + e T ue s)=( 2 I V T V)v, µ s(e T s vet s (2 I V T V ) V T e + e T uet s (2 I V T V ) e s )=e T s v. Similarly, we fid µ s(e T ue T ( 2 I VV T ) Ve s +e T s ve T ( 2 I VV T ) e )=e T u. Notig that e T s ( 2 I V T V ) V T e = e T ( 2 I VV T ) Ve s,wearrive at the 2 2 system µ ( e T s ( 2 I VV T ) Ve s e T (2 VV T ) e e T s (2 V T V ) e s e T (2 I VV T ) Ve s Sice )( ) e T u e T s v = ( ) e T u e T s v 0, it is a eigevector of the above matrix. Thus (µ se T (2 I VV T ) Ve s ) 2 which implies ( ) e T u e T s v. = 2 e T ( 2 I VV T ) e e T s( 2 I V T V) e s µ 2 s, 2 = µ 2 s et (I VV T ) e 2 e T s(i V T V) e 2 s ( µ s e T 2 (I. VV T ) Ve 2 s ) 2 For the followig K will deote a geeric positive costat covergig to a costat ot depedig o the realizatio. Notice µ s e T 2 (I VV T ) Ve 2 s K.Write µ s e T 2 (I 2 VVT ) Ve s = µ s X 2 + Y, where X is defied i Theorem.3, ad

7 SPECTRAL RADII OF RANDOM MATRICES 53 Notice Y K 3/2. Y = µ s 2 e T VVT (I 2VVT ) Ve s. Write e T (I VV T ) e 2 =+ X(),whereX () = et VV T e + Y (), Y () = e T 3 (VVT ) 2 (I VV T ) e 2,ade T s (I V T V) e 2 s = + X(2),whereX (2) = et s V T Ve s +Y (2), Y (2) = e T 3 s (V T V ) 2 (I V T V ) e 2 s. Notice X (),X (2),Y (),Y (2) are o-egative, with max(x (),X () ) K,admax(Y (),Y () ) K For all sufficietly large, we have. = µ s ( + X () )( + X (2) ( µ s X 2 + Y ) Usig +x ( + 2 x) 8 x2 for x 0, x ( + x) 2x2 for x, ad arguig i the same maer i the proof of Theorem 2.2, we fid for all ad all realizatios = µ s + 2 Write µ s 2 (e T VV T e +e s V T Ve s )+ ). ( µ s ) 2 X + Z. e T VV T e +e s V T Ve s =( + s ) ij v 2 ij + A, where A = v ij v ij + v ij v ij. s i i j From the cetral limit theorem { s (vij 2 σ2 )} is tight, ad sice E(A2 )=2( ( )s 2 ij j j i + (s ) ), we fid { s A } to be tight. Therefore = µ s + 2 σ2 µ s( + s) 2 + ( µ s ) 2 X + Z.

8 532 SILVERSTEIN Sice = µ s + Z, we fid ( µ s ) 2 = Z ad µ s( + s) 2 ( ) s µ s + = Z. This completes the proof of Theorem.3. ACKNOWLEDGMENTS The author would like to thak the referees for their helpful suggestios. This paper was writte while visitig the Ceter for Stochastic Processes, Uiversity of North Carolia at Chapel Hill, ad supported i part by NSF ad AFOSR Grat F J-054. REFERENCES [] Bai, Z.D., Silverstei, J.W., ad Yi, Y.Q. A ote o the largest eigevalue of a large dimesioal sample covariace matrix. Joural of Multivariate Aalysis 26(2) (988), pp [2] Füredi, Z. ad Komlós, J. The eigevalues of radom symmetric matrices. Combiatorica 3 (98), pp [3] Hor, R.A. ad Johso, C.R. Matrix Aalysis Cambridge Uiv. Press, 985. [4] Yi, Y.Q., Bai, Z.D., ad Krishaiah, P.R. O limit of the largest eigevalue of the large dimesioal sample covariace matrix Probab. Theory Related Fields 78 (988), pp