The Matrix Analog of the Kneser-Süss Inequality

Transcription

1 Proceedigs of the 9th WSEAS Iteratioal Coferece o Applied Mathematics Istabul Turkey May pp ) The Matrix Aalog of the Keser-Süss Iequality PORAMATE TOM) PRANAYANUNTANA PATCHARIN HEMCHOTE 2 PRAIBOON PANTARAGPHONG 2 Departmet of Cotrol Egieerig Faculty of Egieerig Kig Mogkut s Istitute of Techology Ladkrabag KMITL) 3 Moo 2 Chalogkrug Rd Ladkrabag Bagkok 0520 THAILAND Tel: 662) Fax: 662) kpporama@kmitlacth ppraaya@polyedu 2 Departmet of Mathematics ad Computer Sceices Faculty of Scieces Kig Mogkut s Istitute of Techology Ladkrabag KMITL) 3 Moo 2 Chalogkrug Rd Ladkrabag Bagkok 0520 THAILAND Tel: 66) khpatcha@kmitlacth kppraibo@kmitlacth Abstract: The Bru-Mikowski theory is a core part of covex geometry At its foudatio lies the Mikowski additio of covex bodies which led to the defiitio of mixed volume of covex bodies ad to various otios ad iequalities i covex geometry Various matrix aalogs of these otios ad iequalities have bee well kow for a cetury We preset a few ew aalogs The major theorem preseted here is the matrix aalog of the Keser-Süss iequality Key-Words: Elliptic Bru-Mikowski theory Mikowski iequality Bru-Mikowski iequality Keser- Süss iequality Mikowski s determiat iequality Blaschke summatio Matrix Blaschke summatio Mixed determiat Matrix Keser-Süss iequality Itroductio The Bru-Mikowski theory is a core part of covex geometry At its foudatio lies the Mikowski additio of covex bodies which led to the defiitio of mixed volume of covex bodies ad implicitly to the famous Bru-Mikowski iequality The latter dates back to 887 Sice the it has led to various otios ad a series of iequalities i covex geometry Various matrix aalogs of these otios ad iequalities have bee well kow for over a cetury ad have bee widely use i mathematical ad egieerig applicatios Our purpose here is to develop a equivalet series of iequalities for positive defiite symmetric matrices 2 Materials ad Methods 2 Mixed Determiat ad Cofactors A well kow matrix aalog of the covex geometry otio of mixed volume is called mixed determiat Its defiitio is quoted here as follows: Defiitio Mixed Determiat a []) Let A A r be symmetric matrices λ λ r be positive scalars The the determiat of λ A + + λ r A r ca be writte as Dλ A + + λ r A r ) λ i λ i DA i A i ) where the sum is take over all -tuples of positive itegers i i ) whose etries do ot exceed r The coefficiet DA i A i ) with A ik k from the set {A A r } is called the mixed determiat of the matrices A i A i a The authors choose to quote this defiitio of mixed determiat i a way aalogous to the defiitio of mixed volume i covex geometry [2 3]

2 Proceedigs of the 9th WSEAS Iteratioal Coferece o Applied Mathematics Istabul Turkey May pp ) Properties of Mixed Determiats: Let A A A B ad B be matrices λ λ be positive scalars DA A B) DA A B A) 2 DA B A A) 2 DB A A) I fact the mixed determiat is symmetric i its argumets so i a larger geerality oe has: ) DA A B B) k k DB B A A) k k We use the otatio DA k; B k) to represet ay of DA A B B) DB B k k k A A) i ) k 2 2) Dλ A λ A ) λ λ DA A ) 3 3) DA A B + B ) DA A B) I particular + DA A B ) DA A B + B ) DA A B) + DA A B ) The properties i 2) ad 3) follow from the - liearity of the mixed determiat remark: Remark 2 []) A mixed determiat DA A 2 A ) of matrices A A 2 A ca be regarded as the arithmetic mea of the determiats of all possible matrices which have exactly oe row from the correspodig rows of A A 2 A Defiitio 3 Cofactor Matrix []) The cofactor matrix CA of a matrix A is the traspose of the well kow classical adjoit of A thus it is defied by 5) CA) ij : ) i+j DAi j)) where Ai j) deotes the ) ) matrix obtaied by deletig the i-th row ad the j-th colum of the matrix A We use a similar otatio i matrix theory to represet a aalog of the mixed volume V K L) where K ad L are covex bodies as follows: Defiitio 4 []) D A B) is the followig mixed determiat of matrices A ad B: 6) D A B): DA A B) 22 Matrix Versio of Blaschke Summatio ad Matrix Aalogs of Mixed Volume betwee Two Covex Bodies Learig the properties of the otio of Blaschke summatio of covex bodies i covex geometry we itroduce its aalog i matrix theory as follows: Defiitio 5 Matrix Blaschke Summatio []) The Blaschke Summatio of the matrices A ad B deoted by A + B is defied as the matrix whose cofactor matrix is the sum of the cofactor matrices of A ad B; that is it satisfies the followig equality: 7) CA + B) CA + CB Oe ca show that for matrices A ad B: 4) a b DA A B) a a of which the geeralizatio gives a alterative defiitio of the mixed determiat as i the followig b a Theorem 6 []) Let A [a ij ] B [b ij ] If A B : ij a ij b ij the for ay positive scalar 8) DA + B) DA) D A B) CA B lim 0

3 Proceedigs of the 9th WSEAS Iteratioal Coferece o Applied Mathematics Istabul Turkey May pp ) It is atural to regard the product CA B see the proof i [] as a equivalet of the mixed volume betwee two covex bodies A ad B The previous theorem was proved by the asymptotic expasio of the determiat of A + B which is similar to the Steier s polyomial for the volume of A + B where A ad B are covex bodies Oe ca easily show that DA + B) ca be expaded as see []) 9) DA + B) { i0 ) / ) DCA i; CB i)} i Also DA + B) ca be expaded as 0) { ) / ) DA + B) i DCA i; CB i)} i i0 where B / ) B The as is close to 0 we get { DA + B) D A)+DCA ; CB)} / ) ad we have the liear approximatio ) DCA ; CB) DA + B) DA) + D 2 A) Therefore we have the followig equality: Theorem 7 []) Let A B be positive defiite symmetric matrices be a positive scalar The 2) CB A lim 0 where B / ) B DA + B) DA) 23 The Matrix Aalogs of the Bru- Mikowski the Mikowski the Keser- Süss Iequalities The followig theorem is a well kow iequality proved by Mikowski Theorem 8 Mikowski the Bru-Mikowski iequality [ ]) Let A B be positive defiite symmetric matrices The 3) DA + B) / DA) / + DB) / with equality if ad oly if A cb It is called Mikowski s determiat iequality [4 6 8] ad is a matrix aalog of the Bru-Mikowski iequality i covex geometry Ad here are couple of others Theorem 9 Matrix aalog of the Mikowski iequality a []) Let A B be positive defiite symmetric matrices The 4) D A B) DA) DB) with equality if ad oly if A cb Proof Usig AM-GM iequality: tr Q DQ) / for ay matrix Q with positive eigevalues ad A B : a ij b ij it ca be easily proved ij that for positive defiite symmetric matrices A ad B 5) trab) A B DA) / DB) / ad the equality holds if ad oly if AB ci or A is a multiple of B ; that is A cb Note that the eigevalues of the product of two positive defiite matrices are positive sice λab) λa /2 BA /2 ) The it follows directly from 5) that CA B DCA) DB) DA) DB) ad equality holds if ad oly if c A CA c 2 B or A cb where c c 2 c are costats This iequality is a matrix versio of the Mikowski iequality i covex geometry It ca also be show that the aalog of the Bru-Mikowski iequality 3) is equivalet to the aalog of the Mikowski iequality 4) First we shows 4) implies 3) For ay positive defiite symmetric matrix Q it follows from 4) that 6) CQ Q DQ) )/ DQ) / DCQ) / DQ) / Lettig Q A + B where A B are positive defiite symmetric matrices we have DA + B) / CQ A + B) DCQ) / CQ A DCQ) + CQ B / DCQ) / CQ A DQ) + CQ B DQ) DA) / + DB) / The last iequality follows from 4) This cocludes that 4) implies 3) We will ow show 3) implies 4) By 3) ad with beig a positive scalar we have a Despite lackig of referece literature the authors believe that this theorem is a well kow theorem i matrix theory

4 Proceedigs of the 9th WSEAS Iteratioal Coferece o Applied Mathematics Istabul Turkey May pp ) DA + B) DA) DA + B) / ) DA) DA) / + DB) /) DA) DA) / + DB) /) DA) [ ) DA) + DA) )/ DB) / ) ) + DA) 2)/ 2 DB) 2/ + 2 ] DA) ad as approaches 0 we ifer that ) DA + B) DA) lim DA) DB) 0 which is or CA B DA) DB) D A B) DA) DB) This cocludes the proof that 3) implies 4) Theorem 0 Matrix aalog of the Keser-Süss iequality []) Let A B be positive defiite symmetric matrices The 7) DA + B) DA) with equality if ad oly if A cb + DB) Proof To prove this matrix versio of Kesser- Süss iequality it suffices to show that it is equivalet to the aalog of the Bru-Mikowski iequality 3) Usig 3) we have 8) DA + B) DCA + CB) / DCA) / + DCB) / DA) This shows that 3) implies 7) + DB) Oe ca easily verifies that a matrix A is positive defiite symmetric if ad oly if its cofactor matrix CA is a positive defiite symmetric Let X CA Y CB Sice A ad B are positive defiite symmetric the so are X ad Y Usig the defiitio of Blaschke additio ad 7) we obtai 9) DX + Y ) / DCA + CB) / DCA + B)) / DA + B) DA) + DB) DCA) / + DCB) / DX) / + DY ) / This shows that 7) implies 3) ad the theorem is proved The last iequality was ukow i matrix theory Oe may recogize the equivalet of this iequality i covex geometry where volumes replace the determiats ad covex bodies replace positive defiite symmetric matrices The covexity versio of the last two theorems are give i Appedix A 3 Coclusio The matrix Blaschke summatio ad the AM-GM iequality tr Q DQ)/ as i the proof of Theorem 9 play importat roles i the derivatio of matrix aalogs of otios ad iequalities i covex geometry These aalogs look very similar to their covex geometry versio oes The author believes that a plethora of other matrix iequalities ca be obtaied by choosig strategic positive defiite matrices Q i the AM-GM iequality 4 Ackowledgemets The first author dedicates this paper to Associate Professor Dr Chadi Shah Sep Apr who was like a mother to him He wishes to thak Professor Erwi Lutwak for some very helpful ad ispirig coversatios ad correspodece) o the subject of this article Thaks DrAlia Stacu for her may excellet suggestios for improvig the origial mauscript We thak Professor Tawil Paugma Dea of the Faculty of Egieerig KMITL for allocatig fuds for this research We also thak their Faculty of Egieerig for the resources provided

5 Proceedigs of the 9th WSEAS Iteratioal Coferece o Applied Mathematics Istabul Turkey May pp ) Appedix A The Bru-Mikowski Iequality the Mikowski Iequality ad the Keser-Süss Iequality i Covex Geometry Theorem The Bru-Mikowski iequality [2 3 8]) Let K L be covex bodies i R The 20) V K + L) / V K) / + V L) / with equality if ad oly if K ad L are homothetic The theorem ow amed after Bru ad Mikowski was discovered for dimesios 3) by Bru ) [9 0] Its importace was recogized by Mikowski who gave a aalytic proof for the - dimesioal case Mikowski 90 []) ad characterized the equality case; for the latter see also Bru 894) [2] Theorem 2 The Mikowski iequality [2 3 8]) Let K L be covex bodies i R The 2) V K L) V K) V L) with equality if ad oly if K ad L are homothetic [8] R Webster Covexity Oxford Uiversity Press New York 994 [9] H Bru Über Ovale ud Eifläche PhD thesis Dissertatio Müche 887 [0] H Bru Über Curve ohe Wedepukte Habilitatiosschrift Müche 889 [] H Mikowski Geometrie der Zahle Teuber Leipzig 90 [2] H Bru Referat über eie Arbeit: Exacte Grudlage für eie Theorie der Ovale S-B Bayer Akad Wiss 894 pp 93 Theorem 3 The Keser-Süss iequality [3]) Let K L be covex bodies i R The 22) V K + L) V K) + V L) with equality if ad oly if K ad L are homothetic Refereces: [] P Praayautaa Elliptic Bru-Mikowski Theory PhD thesis Dissertatio Polytechic Uiversity Brookly New York August 2002 Jue 2003) [2] R Scheider Covex Bodies: The Bru-Mikowski Theory Cambridge Uiversity Press New York 993 [3] E Lutwak Volume of mixed bodies Trasactios of The America Mathematical Society April 986 pp [4] R A Hor ad C R Johso Matrix Aalysis Cambridge Uiversity Press New York 985 [5] G P Egorychev Mixed Discrimiats ad Parallel Additio Soviet Math Dokl vol pp [6] M Marcus ad H Mic A Survey of Matrix Theory ad Matrix Iequalities Dover Publicatios New York 964 [7] V V Prasolov Problems ad Theorems i Liear Algebra volume 34 America Mathematical Society Uited States 994