ON THE ELEMENTARY SYMMETRIC FUNCTIONS OF A SUM OF MATRICES

Transcription

1 Joural of lgebra, umber Theory: dvaces ad pplcatos Volume, umber, 9, Pages 99- O THE ELEMETRY YMMETRIC FUCTIO OF UM OF MTRICE R.. COT-TO Departmet of Mathematcs Uversty of Calfora ata Barbara, C 96 U... e-mal: rscosa@gmal.com bstract Ofte mathematcs, t s useful to summarze a multvarate pheomeo wth a sgle umber. I fact, the determat - whch s deoted by det - s oe of the smplest cases ad may of ts propertes are very well-ow. For stace, the determat s a multplcatve fucto,.e., det( B det det B,, B M, ad t s a multlear fucto, but t s ot, geeral, a addtve fucto,.e., det ( B det det B. other terestg scalar fucto the Matrx alyss s the characterstc polyomal. I fact, gve a square matrx, the coeffcets of ts characterstc polyomal χ ( t : det( ti are, up to a sg, the elemetary symmetrc fuctos assocated wth the egevalues of. I the preset paper, we preset ew expressos related to the elemetary symmetrc fuctos of sum of matrces. The ma motvato of ths mauscrpt s try to fd ew propertes to probe the followg coecture. Besss-Moussa-Vlla coecture: [, 4] The polyomal p( t : Tr(( tb R[], t has oly oegatve coeffcets wheever, B Mr are postve semdefte matrces. Mathematcs ubect Classfcato: Prmary C, 5E5, P8. Keywords ad phrases: elemetary symmetrc fucto, Hermta matrx, determat. Receved Jue, 9 m 9 cetfc dvaces Publshers

2 R.. COT-TO Moreover, some umercal evdeces ad the ewto-grard formulas suggested to us to cosder a more geeral coecture that wll be cosdered a further mauscrpt. Postvty Coecture : m The polyomal (( tb R[], t has oly oegatve coeffcets wheever, B M r are postve semdefte matrces for every,, K, r. It s clear that the BMV coecture s a partcular case of the postvty coecture for, sce Tr.. Itroducto Deote by M m,, the set of m matrces over a arbtrary feld F ad by M the set M. Determats are mathematcal obects that, are very useful the matrx aalyss. I fact, the determat of a matrx M, ca be preseted two mportat, apparetly dfferet, but equvalet ways. The frst oe s the Laplace Expaso: If [ ] a, M, the ad assumg that the determat s defed over det ( ( a, det(, ( a, det(,, ( where, M deotes the submatrx of resultg from the deleto of row ad colum. where The secod way s the lteratg um: det ( sg( σ a, σ( a, σ( L a, σ(, ( σ P P s the set of all permutatos of {,,, }, deotes the sg of the permutato σ. K ad sg ( σ

3 O THE ELEMETRY YMMETRIC FUCTIO Remar.. otce that wth these deftos t s clear that the determat s a multlear fucto. I the preset paper, we preset a closed expresso for det( L,,, K, M, where, terms of the sum of aother determats volvg the matrces,, K,. Defto.. Let M m,. For ay dex sets α, β, wth α {, K, m}, β {, K, }, ad α β, we deote the submatrx that les the rows of dexed by α ad the colums dexed by β as ( α, β. For example ({ }, {, } [ 9]. O the other had ad tag to accout some propertes of the determat t s well-ow that the characterstc polyomal of a gve square matrx ca be wrtte as χ ( t det( ti t ( t L ( (, where I M s the detty, ad ( s the elemetary symmetrc fucto assocated to the matrx,,, K,. I fact, by the secod way as we have defed the determat,.e., the alteratg sum, t s straghtforward that ( det( ( α, α,,, K,. α ( I coecto wth the elemetary symmetrc fuctos, we preset ew equaltes related to these fuctos, gvg explct expressos for ( B ad (, for ay B M. B, The structure of ths paper s the followg: I ecto, we preset some results related wth the determat of sum of matrces, whose proof

4 R.. COT-TO s gve ppedx. I ecto, we obta the values of ( B ad ( B by usg the defto of the elemetary symmetrc fuctos of a matrx, I ecto 4, we prove the same dettes ad also we obta 4 ( B by usg the ewto-grard dettes, where ad B are two geerc -by- matrces.. The Determat of a um of Matrces Let be a postve teger ad let us cosder the -tuple of -by- matrces (,, L,. : We defe ( as the set of all possble formal sums of matrces of, where each,, K,, appears at most oce. Remar.. ote that WLG we ca add the ull matrx,, (. The followg result wll be useful for further results Theorem.. Gve M ad a teger, wth. For ay -tuple (,, L,, M,, K,, the followg relato holds: ( det Ω ( Ω Ω, (4 uderstadg that summads, ad that Ω meas that Ω s a formal sum wth Ω meas that s a summad Ω. Remar.. The detty (4 ca be rewrtte as x, K, x x ( L x det. x (5 Chapma proves [7] the case of ths. But hs argumet wors as well ths geeralzed form; the determat s a polyomal of degree less tha the varables x, K, x ad ths alteratg sum must

5 O THE ELEMETRY YMMETRIC FUCTIO vash as see by applyg to ay moomal of degree less tha. lteratvely (5 follows by subtractg the case of Chapmas detty from the case. For stace, f we set (4 ad,,, 4 M,.e., 4, the det ( 4 det ( det ( 4 Ths result has very terestg cosequeces. det ( 4 det ( 4 det ( det ( det ( 4 det ( det ( 4 det ( 4 det ( det ( det ( det( 4. Corollary.4. Uder the codtos of Theorem.. For ay dex sets α, β {,, K, } of sze τ, τ, the followg relato holds: ( det α, β Ω ( Ω ( (, α β. Ω (6 The proof follows from Theorem. replacg by ( α, β ad tag to accout that ( α, β M τ ad τ. O the other had, f we combe the above result ad ( we obta that: Corollary.5. Uder the codtos of Theorem., for ay oegatve teger τ, τ, ( τ Ω ( Ω Ω, (7 where τ ( C s the τ- th elemetary symmetrc fucto of the matrx C.

6 4 R.. COT-TO The proof, aga, s straghtforward tag to accout ( ad that τ. The followg detty s useful to compute τ- th elemetary symmetrc fucto of ay umber of matrces τ. Corollary.6. Uder the codtos of Theorem.. For ay oegatve tegers τ, τ, the followg detty fulflls τ ( (, ( L τ (8 Ω Ω Ω The proof s elemetary ad we leave t for the reader. I fact, Theorem. s optmal wth respect to the rage of,.e., for every postve teger, t s possble to fd -tuples of M such that the equalty (4, gve Theorem., fals. For stace, tag dag( e,,, K,, xe, x R, where { e, e, K, e } s the caocal bass of R, t s straghtforward to chec that ( det Ω ( Ω Ω ( ( x x.. Obtag ( B ad ( B o the ext logcal step s to get closed expressos for the τ- th elemetary symmetrc fuctos of a sum of matrces, wth τ. To do that we wll use the ewto-grard formulas for the elemetary symmetrc fuctos (see, e.g., [5, ubsecto.] ad the defto of such fuctos (. Remar.. ote that f s -by-, the det ( (, so t s eough to obta those dettes for the elemetary symmetrc fuctos ad the apply these to the determat.

7 O THE ELEMETRY YMMETRIC FUCTIO 5 Lemma.. For ay,, M we get ( ( ( ( ( (, (9 ( ( ( ( ( ( ( ( ( ( (. ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (. ( Proof. Let M, be two matrces wth spectra ( σ { },, K ad ( { },,, µ µ σ K respectvely. The ewto-grard formula gves ( (, where ( (. Tr WLG we ca assume dagoal, the by defto of (see ( we get ( ( ( (, b Tr < ( ( ( Tr Tr Tr

8 6 R.. COT-TO ( ( ( Tr( ( Tr( ( Tr ( Tr( Tr( ( ( ( Tr( Tr( Tr(. d for (, f s a dagoal matrx, by defto of (see (, t s straghtforward to get ( Tr( b ( b,, < < < (( (., Tag to accout that ths case the ewto-grard formula produces the detty ( ( ( (, ad the expaso of ( a b c, we obta ( (( Tr( Tr( Tr( Tr( 6 ( Tr( Tr( Tr( Tr( (( (., But we ca assume s a dagoal matrx ad say o, (( ( µ Tr( µ,,, K,., ( ( ( Tr( Tr( Tr(

9 O THE ELEMETRY YMMETRIC FUCTIO 7 Tr( Tr( ( Tr( Tr( Tr ( (, ad hece both relatos, (9 ad (, hold. Moreover ( s a drect cosequece of (. 4. Other Way to Obta ( B, ( B ad 4 ( B We wll start probg ( B usg the ewto-grard dettes: ( B (( B ( B ( ( ( B ( B ( ( B ( B ( ( B ( ( B ( B. We wll apply a aalogous way to obta ( B: ( B (( B (( B ( B ( B ( B ( ( B ( B ( B ( ( ( ( B ( B ( ( B ( B ( B ( ( B ( B ( B (. B If ow we expad ( B, after some smplfcatos t s clear we get the desred detty for ( B. 4.. Obtag ( B 4 s the above examples, the ewto-grard formula gves 4 4 ( B (( B (( B ( B (( B 4 ( B ( B (. B Tag to accout the propertes of the trace, we get

10 8 R.. COT-TO ( B ( 4 ( B 4 ( B (( B ( B 4 ( B ( ( ( ( B ( B ( ( B ( B ( B ( ( B ( B ( B ( ( B ( B ( ( ( B ( B ( ( ( B ( ( B ( B ( ( B( ( ( B ( ( B ( B ( ( B ( B ( B ( ( B (. B pplyg the same techque appled before, we get 4 ( B 4 ( ( B ( B 4 ( B (( B ( B ( ( B ( B ( ( B ( B ( B ( ( B ( B ( B ( ( ( ( B ( ( B ( B ( B( ( ( B ( ( B ( B ( B ( ( ( ( B ( B ( ( ( B ( ( B ( B ( ( B ( B ( B ( ( B ( B ( B( ( ( ( B ( B ( ( B ( B ( B ( ( B (. B fter some smplfcatos applyg the ewto-grard formulas, we get 4 ( B 4 ( 4 ( B 4 ( B 4 ( B (( B 4 4

11 O THE ELEMETRY YMMETRIC FUCTIO 9 4 ( B ( B ( ( B ( B ( B ( ( B ( B ( B( ( ( B ( B ( ( ( B ( B ( B ( B ( B( ( B ( B ( B ( 4 ( ( B 4 ( B ( ( ( B 4 ( ( B 4( B ( B. 4 pplyg the ewto-grard formulas ad after some smplfcatos, we get 4 ( B 4 ( 4 ( B ( B ( B ( B ( B ( ( B ( B ( B ( ( B ( B ( B ( ( B ( ( B ( B ( ( ( B ( ( B ( B ( B ( B. 5. Coclusos ad Outloo We have costructed the d, the rd ad the 4th elemetary symmetrc fuctos of a sum of two matrces but, of course, s smple to see that s possble to compute the τ- th elemetary symmetrc fucto of a sum of -matrces, τ by usg the ewto-grard formulas or by usg the same techque used Lemma. whch, by the way, s too much complcated. Of course, oe of the goals further papers s to fd a closed expresso the geeral case whch for the momet s ot clear although we beleve the Theory of partto of tegers s volved. I fact, by usg the geeralzed Warg s formula [6], for ay s - by- matrx, ad ay oegatve teger m, we get

12 R.. COT-TO m ( ( ( e (, where the coeffcets are gve by l ( ( l ( π l( π l π U ULU ( π ( π! l l (,, m π K ad e ( ( ( m m L ( (. l(, ( m (, K, m ( Remar 5.. partto s a fte sequece (,, K, r of postve tegers decreasg order, where l ( deotes the legth of the partto, ad m ( deotes the umber of parts of equal to. U µ s the partto whose parts are those of ad µ. Tag to accout ths detty, we beleve that we ca obta a aalogous expresso for the τ- th elemetary symmetrc fucto of a sum of matrces. I fact, we expect oe expresso whch appears the elemetary symmetrc fuctos o words of the letters,, K, as oe could see ubsecto 4. for the case 4 (. cowledgemets The author thas the referee for the costructve remars ad the valuable commets. Ths wor has bee supported by Drecto Geeral de Ivestgaco (M-stero de Educaco y Ceca of pa, grat MTM 6--C-. Proof of Theorem. We wll prove by ducto o : ppedx If the matrces are scalars so, for every, det Ω ( Ω ( Ω Ω ( Ω Ω ( m

13 O THE ELEMETRY YMMETRIC FUCTIO ( L, ad hece (4 holds for ad. If we assume that the result holds for, let us gog to prove the detty (4 for. Tag the Laplace expaso through the frst row, we get ( det Ω ( Ω ( ( Ω Ω ( Ω ( (, (, det( ( }.,, Ω Ω { ( By ducto, sce for every,, K,, (, s fxed ad does ot deped o or Ω, we get that the above expresso s equal to ( ( (, det(, (, Ω ( Ω Ω Ω,, Ω ( Ω ( ( (, Ω Ω det( (. ow, f we assume that ay set wth less tha oe elemet has determat equal to zero, we get ( ( (, ( (, ( det ~ Ω ( \{ } ~ Ω ( ( (,, ~ Ω,,,, ~ ~ Ω ( \{ } Ω ~ Ω det( ( (.

14 R.. COT-TO By ducto, sce for every ad, the matrces,, ( M are fxed, thus, ( M s also fxed. Thus we get, ( (,. Moreover, sce, the. Hece, the relato holds. Refereces, [] M. ger, Combatoral Theory, prger-verlag, Orgally publshed 979 as Vol. 4 of the Grudlehre der math., Wssechafte, reprted 997. [] D. Besss, P. Moussa ad M. Vlla, Mootoc covergg varatoal approxmatos to the fuctoal tegrals quatum statstcal mechacs, J. Math. Phys. 6 (975, 8-5. [] R. Hor ad C. Johso, Matrx alyss, Cambrdge Uversty Press (Cambrdge, 985. [4] E. H. Leb ad R. erger, Equvalet forms of the Besss-Moussa-Vlla coecture, J. tatst. Phys. 5( (4, [5] R. éroul, Programmg for Mathematcas, prger-verlag (Berl,. [6] Jag Zheg, O a geeralzato of Warg s formula, dvac. ppl. Math. 9 (997, [7] mer. Math. Mothly 9(7 (, g