ON A CHARACTERIZATION OF THE BEST R, SCALING OF A MATRIX G. H. GOLUB J. M. VARAH STAN-CS OCTOBER 1972

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1 SU326P30-22 ON A CHARACTERIZATION OF THE BEST R, SCAING OF A MATRIX BY G H GOUB J M VARAH STAN-CS OCTOBER 1972 COMPUTER SCIENCE DEPARTMENT School of Humates ad Sceces STANFORD UNIVERSITY

2 f O a Characterzato of the Best &2 Scalg of a Matrx c GH:Golub Computer Scece Departmet Staford Uversty Staford, Calfora, USA ad JM Varah Computer Scece Departmet Uversty of Brtsh Columba Vacouver, BC, Caada Issued jotly as a Techcal Report from Staford Uversty Computer Scece Departmet ad The Uversty of Brtsh Columba Computer Scece Departmet The work of the frst author was supported by NSF ad AEC grats; that of the secod was supported by NRC(Caada) grat #A8240

3 Abstract Ths paper s cocered wth best two-sded scalg of a geeral square matrx, ad partcular wth a certa characterzato of that best scalg: amely that the frst ad last sgular vectors (o left ad rght) of the scaled matrx have compoets of equal modulus Necessty, suffcecy, ad ts relato wth other characterzatos are dscussed The the problem of best scalg for rectagular matrces s troduced ad a cojecture made regardg a possble best scaug cases The cojecture s verfed for some specal

4 1 Itroducto et A be a x osgular matrx+ We are terested the best row ad colum scalg of A the l2 orm; that s m D,E dag Of course ths s equvalet to -- m D,E dag tolow /a WE)) where al(a) 2 a2(a) 2 2 o,(a) > 0 are the sgular values of A I ths paper we wll dscuss the followg useful characterzato of ths best two-sded scalg: let A = U 1 V be the sgular value decomposto of A The A s best scaled the e2 orm f /u(1) 1 r Iu(d l,lv~l'l = Iv'/, -1 t* s for That s, A s best scaled f the frst ad last colums of U ad V have compoets of equal magtude m We refer to ths as the EMC property Ths characterzato has had a terestg hstory: t was to our kowledge frst dscussed by Forsythe ad Straus [3] coecto wth c oe-sded scalg, or equvaletly best symmetrc scalg (DAD) of a postve defte matrx A (For oe-sded scalg, oly oe of U,V s volved the EMC property) They showed suffcecy*of EMC for best oe-sded scalg It was also metoed by Bauer [l] for oe-sded scalg; he also gave a explct represetato of the best t, scalg for matrces A wth A ad A-' havg a checkerboard sg patter More recetly, McCarthy ad Strag [4] have settled the questo of ecessty for oe-sded scalg: for matrces A whch whe best scaed have al ad a dstct, the EMC

5 2 property must hold; however ths s ot always true f al or a s multple eve usg the heret ambguty the sgular vectors, ad they gve couterexamples The EMC property for two-sded scalg was frst brought to our atteto by C awso (see also [6, pg 447) coecto wth the matrx We foud the best e2 scalg by mmzg ul(dae)/u(dae) as a fucto of D,E usg a fucto mmzg procedure Ths gave D = dag(l,&3), E * dag(l/2,l/h), u /U 2 139, 1 ; DM-(-' -;;; ;; ) c I ths paper, we dscuss the EMC property for best two-sded scalg ad how t s related to the Bauer represetato for checkerboard matrces The we dscuss the problem of best scalg for a rectagular matrx We ed ths troducto wth a warg: although these best scalgs are attractve ad theoretcally terestg, t may be qute mproper to scalea partcular problem ths way; ths ca cause accurate data ad umportat varables to assume too much fluece Such s the case for example solvg ll-posed problems usg the sgular value decomposto be [51> Normally several of the equatos are gored ad a reasoable soluto s costructed solvg the remag oes; however best" scalg ca cause the whole matrx to bec,ome qute well-codtoed, wth ts (well-determed) soluto bearg o relato to the soluto of the orgal problem

6 2 Aspects of the EMC Property 3 Frst we show the suffcecy of the EMC property for best two-sded scalg The proof s a slght exteso o Forgythe ad Straus [3] Theorem 21: et A be a x osgular matrx The A s best scaled the l2 orm wth respect to dagoal scalgs DAE f the EMC property holds Proof: We have for ay osgular dagoal D,E, - cod2(dae) = u /o'-- = lmx I Psq*r,s t b m -I IP'ACrl td-'pl21te-'ql I2 rthl -1-1 lb r~~2~~e sl12 cod2(a) ($dte-v( )) (v(l>t~w2v(l>) I 112 I t where u(l),ut),v(,vc) are the approprate sgular vectors of A Now f Id")> I - I (u(l)> I ad d$ = I#')),1 for - l,,, e f the EMC property holds, for all D,E QED the term square brackets s 1 ad cod2(a) s cod2(dae) For the EMC property to be also ecessary for best two-sded &, scalg, we must show the exstece of a D,E wth DAE havg the EMC property However as we metoed earler, McCarthy ad Strag [4] gave examples of oe-sded best scaled matrces for whch thexorrespodg oe-sded EMC property faled

7 to hold These examples hoever had multple Q, or o- best scaled form; for matrces wth dstct ol ad o best scaled form, they showed that EMC was attaable From ths we easly obta: Corollary 22: et A have dstct u1 ad u best scaled form; the the EMC property s ecessary ad suffcet for best two-sded e2 scalg, Thus the exstece of a EMC scalg s assured wth ths restrcto A u c of dstct extreme sgular values Of course t eed ot be uque: for example f A has a specal symmetry so that PAQ=A for P, Q permutato matrces, the f DAE s best scaled, so s (PDPT)A(QTEQ) (Ths s P(DAE)Q wth sgular value decomposto (PU)c(VTQ) ad ths has EMC f U 1 VT does) b c Now we dscuss the relato betwee EMC ad Bauer's characterzato for best e2 scalg of a real rreducble checkerboard matrx A We must also assume, although t ormally follows from the rreducblty of A, that IAl la-1l s la 1l IAl s IAl latl, latl IAl e are rreducble Recall the Bauer characterzato (see [l]): f A, A-l have checkerboard sg patters, that s f there exst dagoal orthogoal matrces, J1, J2, J 3, J4 so that JlAJ2 = IA/ z 0 ad J3d1J 4 - /Agll z 0, ad f we let y(l), x(l) be the left ad rght Perro egevectors of A/ IAN11 (ad smlarly y (2), xc2' for 1~~~1 IA/>, the the best C, scalg DAE for A s gve by d 2 ID p/ X%(l),

8 2 e 'x (Because of the rreducblty, the Perro vectors have postve compoets) Thus A s best scaled f the left ad rght Perro vectors of IA/ l&l ad IA -1 ItAl are equal 5 But such a matrx A satsfes the codtos of Corollary 22, so thk above must be equvalet to the EMC property We expad o ths as follows: Theorem 23: - et A be a real rreducble matrx wth a checkerboard sg patter Suppose IAI=J~AJ~ 1s -1-1, IA I'J3A J4 ad let A = U 1 VT be ts sgular value decomposto, A- I () Suppose the Perro vector of 1, 4 A-l, ad Iv(') I s the left ad rght Perro vector of IA~~IIAI () Suppose the EMC property holds The lu(l) I s the left ad rght left ad rght Perro vectors of IAl IA -1 I are equal (call Tt u), ad smlarly for IA-l/ IAl (call t v) The u(l) - J1u, u(~) = J u, 4 p - J2v, I+) = J v 3 Proof: () We have IAl - JlAJ2 - (J U) 1 (VTJ ) ad ths must be the sgular value decomposto for IA/ Hect J1u@) >'O, J2v(l) > 0 (postve because of the rreducblty of A) Smlarly IA-'1-1 -JA 3 - (J3V)lm1(UTJ4) ad J4 we must have J v Cd > 0, J4u() > 0 Now the EMC property ad orthogoalty of the- ~u(~)i;~v(~)i gves J4u( ) =J1u(l) J1u( ) =J4u(l) J3v( ) =J2v(l) J2v( ) =J3v(') Jlu(') IJ u(l) 4 J2v(l) 1J v(l) 3 (1) NOW IA/ IA-~/ = ~~ IJ 1 (VJ,J~V~ ~-'-u~j 4 - ~~ u (1 Q 1-l) U~J 4 Cosder Q; t s orthogoal ad symmetrc, ad from (1) we see that

9 6 Ql-Ql=l ad the rest of the frst ad last rows ad colums of Q are zero Thus 'AIlA-+(J,u(') )'J1 U (CQC-l) UT J4(J4~()) =J1 U (~Q~% -- -J1 U ()e ul (J u(l)) 1 1 So J1u(l) = 'u(l) 1 s the uque postve rght Perro egevector of -1 IAl IA I correspodg to the egevalue u /a A smlar computato shows 1 ' t s also the left Perro vector kewse, J2v (1) p '$) Ica be show to be the left ad rght Perro vector for lawlllal t () If the hypothess of () holds, the from Bauer [l] we have that I IAl s al/o, the spectral radus of IAIIA -1 1 ad IA -1 Thus IA'lA"' u - Q u1 u, whch gves 1-1 uj3a J4u - u J A -1 J u 12 1 Now let J4u - 1 C u (0 l above ca be wrtte, J1u - 1 B l u('), a a 1 vc) = (J3J2) al 1 -v % () 1 u 1 % wthfo2=lb211 l 0 The the

10 7 Now take l2 orms: ad equalty must hold, mplyg that a - 1, zero, gvg J4u - u W, Ju=P 1 J v = IT"), J3v = vc) QED 2 B 1-1 wth the other compoets By a smlar argumet, oe ca show We should also remark that the equvalece of these two characterzatos ca be used to check the accuracy of A-l have checkerboard sg patters ca compute the best scalg va the A-l whe t s kow that both A ad -1 For a gve A ad computed A, oe Perro vectors of IAl IA-l1 ad I~'11 IAI; the oe ca test--the EMC crtero o the sgular vectors of the scaled matrx 3 Best Scalg for Rectagular Matrces et A be m x wth m > ad rak The we ca stll ask for the best scaled DAE the sese of mmzg al/a(dae) It s clear that for the best scalg o the rght, the EMC property o V s stll suffcet, T sce A A s stll a osgular x matrx ad the Forsythe-Straus argumet stll holds However ths s ot the case for scalg o the left, sce partcular we could take ay learly depedet rows of A ad best scale the resultg x matrx; ths wll the have the EMC property (assumg ul ad u are dstct) but wll ot ecessarly gve the best scalg for A There are fact (:) such choces of x submatrces, so a leadg coteder for the best scaled A would be that x submatrx whch whe best scaled gves the mmal codto Ths leads to the trgug Cojecture: There exsts a x submatrx of A whch, whe best scaled, gves the best scalg for A also

11 It would be better to say oe of the best scalgs because t s ot 8 ecessarly uque We caot prove ths geeral, oly some specal ' cases whch we dscuss below several examples Case I: We have also verfed t umercally o B A - (,) where B s x, osgular, ad F T F s dagoal The ATA=BTB + BTFTFB =BT(I + FTF)B -BTGTGB 1 I I- so the ozero sgular values of A ad GB are the same Now f FTF s dagoal, G s dagoal, ad thus the best scalg for A occurs whe GB (or B) s best scaled So oe best scalg for A s DAE - (Dl:E) where DlBE s best scaled However ths s ot ecessarly uque: let B be best scaled, ad cosder DlBE DAE- ( D2FBE 1 The (DAE)T(DAE) = EBT(D12+FTD22F)BE=(GBE)T(GBE) Now f F s such that FTD2*F s dagoal for all D2 dagoal (eg f F 4 has at most oe ozero elemet each row ad colum), the G s also dagoal for all choces of Dl ad D2 ad the best scalg of A occurs for E = Iad ay Dl, D2 such that G - I (sce B Is best scaled) must have That s, we I 2 + FTD *F - I Dl 2 Of course ths wll occur for Dl - I, D2 = 0, but there ca be may other solutos Note also that f B s a orthogoal matrx, a best scalg s certaly obtaed wth Dl = I, D2 '= 0, o matter what F s

12 9 CaseII: =* We have A = (u1 U ad we seek m cod2(b-dab) DSE, D - dag(dl,,d,), E - dag(el,e2), - f g(w) et BTB = (", ) el Cd u ele2cd "P 2 ele2cd "P 2 e2 2 2 Cd 5 The g(d,e) Jf(D,E) 2 2 where f(d,e) - ('-') +4r 1 - Jf(D,E) (P+d2 \ Sce g s a mootoe fucto of f, we eed oly fd m f(d,e) As a fucto of e = e2/el, we ca wrte f(w) = (a-ye*)* + 4e*$* (a+re2) * where a, B, y are costats -e* - v/a, makg p = s ad thus Ths s mmzed as a fucto of e for T B B = (', z) whch has egevector matrx I P ossessg Wth ths E, the EMC property b 2 f(d) =+ P 2 2 (Cd "&v> (Cd2u2)("d2v2), - cos*8(du,dv) To mmze ths, we eed to exame three cases (0 someu orv =O f(d) - 0 for ay choce of the other d j' Suppose u = 0, v # 0 Takg d + 00 gves If u = v - 0, the problem

13 reduces to oe of lower dmeso So assume all u,v#o 10 () $1, $1 ot all of the same sg Suppose ul 3 0, u2 > 0, vl > < 0 for example The we ca make (Du) I (Dv) ad f(d) - 0 by choosg dl A*---- Ju v 11 J-u v 22 s d = 0, # 1,2 If r = ul/vl, R = - u2/v2, ths gves e 2 arrad I best B - DAE - Jr JR 0 JR - Jr 0 I t 6 -ta egevector matrx wth the EMC property () u > 0, v > 0 for all The from a result of Cassels (see Beckebach ad Bellma [2, p 45]), we have c ma f(d) - 4rR II 4 D (HR)* 2+;+g I d r where r - m u /v = um/vm bay) ad R = 1 u/v IJ tqvm The correspodg Dhadm 1 B Ju v mm % = 1,d "O,jmM 2 Ths gves e = rr ad J VM best B = DAE J 0 /r 0 \ JR 1 mth row JR Jr 0 0 \ 6 Mth row

14 I 11 -== I Aga BB T has ts egevector matrx wth the EMC property Fally, oe mght thk that for rectagular matrces wth a checkerboard sg patter,the best scalg could be acheved usg Bauer's algorthm wth A ad A*, the pseudo-verse We gve the followg couterexample: Best scalg: D --dag(l,o,1/2), E - (0' $ The B - DAE - wth cod*(dae) = I Now B' 1/ l/3 > ad Both of these are symmetrc so both have equal left ad rght Perro vectors Thus the Bauer b, scalg leaves B uchaged, f we call O/O = 0 (otce 4 14 s reducble) However f we try to derve B from A usg Bauer's algorthm, t fals: ad ths has spectral radus - p = 362 > 3 - cod2(b) ad rght Perro vectors of IA$IIAI are Moreover the left

15 ' 12, ot optmal We mght also remark that f the cojecture s vald for arbtrary m x matrces, t would dcate the folly of tryg to best scale a rectagular matrx arsg from a least squares problem for example; oly of the observatos would be retaed! Refereces F Bauer, Optmally scaled matrces Num Math 5 (1963), Beckebach ad Bellma, Iequaltes Sprger-Verlag, Berl,1965 GE Forsythe ad EG Strauss O best codtoed matrces Proc AMS 6 (l955), C McCarthy ad G Strag, Optmal codtog of matrces Sam 1 J Num Aal (to appear) '5 JM Varah, O the umercal soluto of ll-codtoed lear systems wth applcatos to ll-posed problems Sam J Num Aal (to appear) 6 GE Forsythe ad CB Moler, Computer Soluto of ear Algebrac Systems Pretce Hall, New York, 1967