ON THE OPTIMALITY OF A CLASS OF H MATRICES

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1 THE PUBLISHING HOUSE PROCEEDINGS OF THE ROMANIAN ACADEMY Series A OF THE ROMANIAN ACADEMY Volume 6 Number /5 pp. - ON THE OPTIMALITY OF A CLASS OF H MATRICES by Ricar GABRIEL Hartstraße Karlsrue Germay We erive a ecessary coitio for optimality of H- matrices i C wic are efie i sectio. We sow tat for tis coitio is also sufficiet a we state a coecture for >.. H MATRICES Accorig to [] a H matrix is a matrix wic ca be writte as Reccomee bymarius IOSIFESCU member of te Romaia Acaemy GD+DH-HD (.) were: D iag(! ) is a iagoal matrix a H is ermitea: H H. A equivalet otatio for G is GD+DH-HD ; ( )] were ( ) are te off-iagoal etries. For a matrix A C let us cosier te optimisatio problems () max Diag TAT ; TT E (i.e. T is uitary) a () mi A Z ; ZZ Z Z (i.e. Z is ormal) G [ (.) Here Diag( A ) iag ( a a! a ) a A a is te Frobeius orm of te matrix A. Tese problems ivolve H matrices: if a iagoal matrix D is a statioary poit for problem () of te best ormal approximatio te A must be a H matrix wit Diag ( A) D(Teorem i []). O te oter a for a H matrix A te ietity matrix E is a statioary poit for problem (); te coverse is ot always true but it ols we E is a seco-orer statioary poit. We oly global extrema are cosiere te problems () a () are always equivalet (Teorem 5 i [] a Teorem i []). If E is a global solutio to () a/or D is a global solutio to () te A must be a H matrix: A G D + DH HD ; i our previous papers we calle it a optimal H matrix. I te case optimality meas (Teorem 5 i []) tat G as te form / ( ) ; ; ( ) G a is equivalet to te seco-orer statioary poit coitio. (.)

2 H matrices also occur as limit poits of a Jacobi sequece { } Ricar GABRIEL A were te iitial matrix A C is a arbitrary (ormal or o-ormal) matrix (Teorem i [9]). We will ow state te seco orer optimality coitio for problem () we te solutio to te problem is a iagoal matrix D. We cosier te geeral case we ot all iagoal etries of D are istict i.e. for is allowe. Let Clearly te pairs {( ) } [5]): D {( ) } I ; (.4) belog to te set I D. Te seco orer optimality coitio is (Teorem i G z z z + z z ( Z) ( ) l l l l l l l ( ) ID l ( ) ID l ( z l l + z l l ) ( l ) ( z l l + z l l ) ( l ) Z i.e. te essia G ( Z) is positive semi- efiite o z C for all [ ] C. (.5) A H- matrix i G ( ) ( ) ( ) wit. Because C is a matrix of te form. H- MATRICES IN C ( ) ( ) ( ) ; (.)! (.) D coul be viewe as a o trivial maximal egeerate iagoal matrix. Teorem. For a H- matrix (.) coitio (.5) is equivalet to / ; (! ). (.) Proof. First for a H- matrix all etermiats i (.5) are zero a o te oter a so te essia becomes D { ( ) ; } I! (.4)

3 O te optimality of a class of H matrices wit a G ( Z) z + z z + z g z z z z g z z z z g z z z z (! ;! ) (! ;! ) (! ;! ) (! ;! ) g z z z z z z z z ( ) z + z g + ( z! z ; z! z ) z z + z z Associate wit te form g ( z z ; z! z ) ermitea matrix Ω ( ) () T E! except for te positive factor ( ) E E ( ) E Te caracteristic equatio of te matrix is T (.5) (.6) (.7) is te (.8) Ω (.9) T 4 ( x ) ( ) x (.) a te ozero roots are a ; terefore te eigevalues of te matrix Ω ( ) are 4 a [ ( ) times]. (.) Hece te matrix Ω ( ) is positive semi-efiite if a oly if 4 tat is /. I orer to sow tat te coitio / is also sufficiet for global optimality i problem () we soul ivoe Teorem 5 i [8]. For a ormal matrix B let us eote { } KomH ( B) X; X X XB BX (.) a for a ermitiea matrix H let δ (H) be its spectral iameter: δ ( H) ymax ymi were y max a y are respectively te largest a te smallest eigevalues of H. Furter wit mi

4 Ricar GABRIEL 4 { δ H } δ( HB ) if ( H X) ; X Kom ( B) (.) Teorem 5 i [8] states tat a sufficiet coitio for a ormal matrix B to be te best ormal approximatio (i Frobeius orm) for te matrix A B + BH HB is For a H- matrix G let us tae δ ( H B). (.4) ( ) B iag!. (.5) Te every matrix X i Kom ( B) ca be expresse as ece te matrix H x X X X X H X will essetially ave te form x H X X X X (.6) (.7) Now if we cosier te caoical ecompositio of X : te result state will follow from Lemma : We ave X Tiag ( x! x ) T ; TT E x u x x δ δ (.8) X u # x were u T. Te poit ere is tat T u. We will sow tat for tir orer H- matrices G ( (. THIRD ORDER H- MATRICES ) ) te coitio ( + ) ( ) ( ) ; / uiquely ietifies global extrema i problems () a (). To tis aim cosier te caracteristic equatio of te matrix (.) y p y + p y p (.)

5 5 O te optimality of a class of H matrices a te problem mi x H x x (.) x {( y ); ( x x x R } y (.4) max mi ) were we get y max a y mi are te largest a respectively te smallest eigevalues. Puttig q q x x x s x + x + x q xx + xx + xx p p p q s q q x x (.5) (.6) It is ow easy to see tat we ca always coose x x a x suc tat y ymax ymi y (.7) wit y betwee y a y : y y y. Equatio (.) as te roots y y a y s. Besies y a y verify te equatios wic lea to y py + p y + p I y + q II q y x x + q (.8) (.9) From I we get y q. If x x x we ave te solutio y max y. We sall prove tat tis is a global solutio to problem (.4) i.e. tat q for ay oter values x x x. To tis e we sow tat te cotrary assumptio q > leas to a cotraictio. Lemma : If eiter x + x x + x or x + x (.) te q. Proof (for x + x ). If x x te q x ( x + x ) x x (.) Lemma : If q > te sg( x + x ) sg( x + x) sg( x + x) sg( x + x )( x + x)( x + x). (.)

6 Ricar GABRIEL 6 Proof. We ave ( x + x )( x + x ) x + q > ( x + x )( x + x ) x + q > ( x + x )( x + x ) x + q > (.) Lemma 4: Te ietity ols. Proof. Obvious. Suppose ow x a let q q q x + x )( x + x )( x + ) (.4) ( x x x x t x x t x. (.5) From Lemma we ave sg( + t ) sg( + t ) sg( + t) sg( t + t)( + t )( + t). (.6) By multiplyig I (i.8) by s q a euctig II y is elimiate; usig (.5) we obtai R qq q q + x + x ( + t )( + t )( t + t ) x ( + )( + t + t ) x+ t x+ t x a a straigtforwar computatio yiels (.7) From I we obtai x ( + t ) + ( + t). (.8) ( + t )( + t )( t + t ) y q + t + t + ( t t ) x (.9) or by (.8) t ( + t) + t ( + t ) y. (.) ( + t )( + t)( t + t) Now we are reay to prove te followig result Teorem : For a H- matrix (.) te coitio ( + ) / (.) is ecessary a sufficiet for D to be a global extremum i problems () a (). Proof. We oly ave to sow tat a cotraictio occurs we assumig q >. To tis e we relay o te iequality y (ece y ). Let us estimate q s

7 7 O te optimality of a class of H matrices E s y ( + t + t ) x y + ( + t) ( + t + t) t ( + t) ( + t + t) t ( + t)( + t)( t + t) ( + t)( + t)( t + t) ( + t) + t ( + t) + t ( + t)( t + t) + t ( + t)( t + t) + t (.) wic soul be egative. But as ca be see from Lemma a te associate relatios (.6) eac term i te above sum is o egative. Now we oly ave to cosier te special case x we system (.9) taes te form y + x x (.) ( x + x ) y x x. (.4) We ca assume x. Te wit x x x x by elimiatig y i (.) a (.4) we get x +. (.5) ( + ) For y xx te computatio yiels wile for s y we obtai te expressio s y + y (.6) ( + ) ( + ). (.7) + wic soul be egative. But te coitio q x implies ece all terms i (.7) are o egative. Te cotraictio also proves te teorem i tis special case. Coecture. I accorace wit Teorems a oe ca expect tat te coitio ( ) /! is ecessary a sufficiet for D to be a global optimum i problems () a () for a arbitrary H- matrix i C. ACKNOWLEDGEMENT. I woul lie to express my tas to M.Iosifescu for is remars cocerig a previous versio of tis paper a to C. Foias a D. Voiculescu wo itrouce me i 97 to problem () of best ormal approximatio.

8 Ricar GABRIEL 8 REFERENCES. R.L. CAUSEY O closest ormal matrices. Tec. Rep. CS- Dept. Comp. Sci. Stafor R. GABRIEL Matrize mit maximaler Diagoale bei uitärer Similarität. J. Reie Agew. Mat. 7/8 pp R. GABRIEL Optimale H-Matrize mit eiem Bezug zur Grapeteorie. Rev. Roumaie Mat. Pures et Appl. 6 pp R. GABRIEL Matrize mit statioärer Diagoale bei uitäre Similarität. Rev. Roumaie Mat. Pures Appl. 9 pp R. GABRIEL Zu e implizite Optimailtätsbeiguge zweiter Orug eier H-Matrix. Rev. Roumaie Mat. Pures et Appl. pp R. GABRIEl Azieee H-Matrize ie ict optimal si. Rev. Roumaie Mat. Pures et Appl. pp R. GABRIEl Te ormal H-matrices wit coectio to some Jacobi-lie metos. Liear Algebra Appl. 9 pp R-GABRIEl Zur beste ormale Approximatio omplexer Matrize i er Euliisce Norm. Mat. Z. pp R. GABRIEl Jacobi s Meto for ormal Matrices of tir a fourt orer. Rev. Roumai Mat. Pures Appl. 4 pp R. GABRIEL Miimizatio of te Frobeius orm of a complex matrix usig plaar similarities. Applie Numerical Matematics 4 pp C. G. J. JACOBI Über ei leictes Verfare ie i er Teorie er Seularstöruge vorommee Gleicuge umerisc aufzulöse. J. Reie Agew. Mat. pp L. LASZLO Optimal plae rotatios for complex matrices. A. Uiv. Sci. Buapest Sect. Comp. pp L. LASZLO Seco orer optimality coitio for H-matrices. BIT (to appear). 4. A. RUHE O te quaratic covergece of te Jacobi meto for ormal matrices. BIT 7 pp V. V. VOEVODIN H. D. IKKRAMOV Über eie Erweiterug er Jacobisce Metoe. I: Vycisl. Met. Programmirovaie VII pp.6-8. Izat. Moscov. Uiv. Mosow 967. Receive November 4