Math Sci Ltt Vol No 8-87 (0) adamard Exotial al Matrix, Its Eigvalus ad Som Norms İ ad M bula Mathmatical Scics Lttrs Itratioal Joural @ 0 NSP Natural Scics Publishig Cor Dartmt of Mathmatics, aculty of rt ad Scic, Mustafa Kmal Uivrsity, atay, Tury Dartmt of Mathmatics, aculty of rt ad Scic, Siirt Uivrsity, Siirt, Tury * Corrsodig author: -mail: drahmti@gmailcom bstract: I this ar, w study th adamard xotial al matrix of th form W foud i j i, j0 orms, two ur bouds for sctral orm ad igvalus of this matrix ially, w giv a alicatio rlatd adamard ivrs, adamard roduct ad igvalus of this matrix Kywords: al matrix; adamard xotial; adamard roduct; orm; sctral orm; igvalu Itroductio I [], Rams rovd that is ositiv smidfiit whr R b symmtric ad ositiv smidfiit R b almost ositiv (smi) dfiit th rlatd adamard ivrs ad adamard squar root of symtric matrics is ositiv (smi) dfiit Morovr, Rams gav som roofs I [], Sola ad Bozurt foud a ur ad lowr boud of Cauchy-al matrix i th form / ( ( ) ) i, j a i j b whr b 0, a ad b ar ay umbrs ad a/ b is ot itgr form I [3], Sola ad Bozurt dtrmid bouds for th sctral ad / ( g h ( i j )) / ( g h ), 0,,,, i, j i, j whr is dfid by i j (mod ) form orm of Cauchy-al matrics of th I [4], Gügör foud lowr bouds for th sctral orm ad Euclida orm of Cauchy-al matrix i th i, j / ( ( ) ) g i j h I [5], Türm ad Bozurt obtaid a ur bouds for th sctral orm of th Cauchy-al matrics of th form / ( g ( i j) h), whr g / ad h I [6], Nallı studid th adamard xotial GCD matrics of th form E ( i, j), i, j whr ( i, j ) is th gratst commo divisor of i ad j Nallı gav th structur thorm ad calculatd th dtrmiat, trac, ivrs ad dtrmid ur boud for dtrmiat of this matrix
8 İ t al: adamard Exotial al Matrix al matrix is a matrix whr h, ad i, j h, () i j hi j i, j0, i, a matrix of th form h h h h h h h3 h h h h h h h h h 0 3 4 Lt a is a m matrix, th adamard xotial ad adamard ivrs of th matrix is dfid by a ( ), a rsctivly [] Lt a is a m matrix, th trasos of th matrix is m Lt a is a m comlx matrix Th / orm of is dfid by T matrix ad dfid by ( a ji ) a (3) i j If th orm is calld robius or Euclida orm ad showd by Lt b m comlx matrix Th th sctral orm of th matrix is dfid by, i (4) max i whr i umbrs ar igvalus of th matrix ad th matrix Th iquality, btw th robius orm ad th sctral orm is valid [7] is cojugat trasos of th matrix (5) Th sctral radius is ow th maximum of th absolut valus of th igvalus of a matrix That is, for matrix, th sctral radius of is dfid as max i Lt a ad dfid by by ad whr i ar igvalus of th matrix i ad B ( b ) is a m matrics Th, th adamard roduct of ad B is trywis roduct B ( a b ) [] Dfi th maximum colum lgth orm c () ad maximum row lgth orm r () o m matrix ( ) a c ( ) max a max a j j i i m ()
İ t al: adamard Exotial al Matrix 83 r ( ) max a max a i i j j Lt a, B ( b ) ad C ( c ) is a m matrics If C B th C r ( ) c( B) [8] (6) Lt a is a matrix Th ricial - miors of matrix ar dfid by i, i,, i i i a a a a a a ii ii ii ii ii ii dt,,,, i a a a i i i i i i whr i i i ( ) Lt ( a ) is a matrix Th quatio dt I a a a a 0 (7) is calld th charactristic quatio of th matrix Charactristic olyomial of matrix is a moic olyomial ad cofficits of this olyomial ca obtai usig ricial miors by,,, i i i a ( ) i, i,, (8) i s a scial, w ca writ ad a a a tr( ) a ( ) i, i,, i a ( ) i, i,, i ( ) dt( ) Taig hi j i j i () w gt a al matrix 0 3 3 4 ad adamard xotial of this matrix is 3 3 4 (9) (0) It is ow that
84 İ t al: adamard Exotial al Matrix 0 x x x x x x Usig this quality, it ca b writ x x x x x x () x If w ta th drivativ of both sid of quality () d d x x dx dx x thus w gt x ( ) x x ( x ) ( ) x x x x () ( x ) I this ar, w ivstigat orm, sctral orm ad igvalus of i (0) ftr, w giv som rsults for dtrmiat ad sctral radius of this matrix ially, w giv a alicatio rlatd adamard roduct ad adamard ivrs as a thorm Mai Rsults Thorm Lt as i (0) Th th orm of this matrix is / / Proof If w calculat th owr of orm of w gt ( ) ( ) ( ) ( ) ( ) ( ) Usig () ad () w gt If w ta (/ ) th owr of th both-had sid w gt / / (3) Thorm Lt as i (0) Th, followig iqualitis for th sctral orm of
İ t al: adamard Exotial al Matrix 85 ad ar valid T Proof or i (3) w gt th robius orm of by E Usig iquality (5) w gt ad Thorm 3 L as i (0) Th, scod ur boud for th sctral orm of is 44 4 ( )( ) ( ) Proof W ca writ 3 3 4 B Th, usig () w gt r ( ) ( ) ad 4 c ( B) ( ) 44 rom (6) a ur boud is foud by 44 4 ( )( ) r ( ) c( B) ( ) Thorm 4 Lt a al matrix as i (0) ad,,, ar igvalus of Th ad 0
86 İ t al: adamard Exotial al Matrix 4 Proof Lt charactristic quatio of is dt I a a a a 0 Now w ca calculat th cofficits a, a,, a usig formula (8) It is s that a tr 4 ( ) ( ) Bcaus of th row ( ) of is qual to 0 Thus w ca say asily that Th a a3 a 0 4 I dt ( ) ad w ca writ 4 ( ( )) 0 4 0, Thus roof is comltd Coclusio 5 Th sctral radius of ad 4 dt( ) 0 Thorm 6 Lt is multil of th first row, vry a al matrix as i (0) Th th igvalus of th matrix ( ) 0, Proof If w writ th adamard ivrs of ( ) 3 ( ) 3 4 ( ) ( ) ( ) () If w writ th adamard roduct of ( ) ( ) B Lt charactristic quatio of B is, it is asily s that ad dt I B b b b b 0 If w calculat b, b,, b w s that immdiatly ad b tr( B) b b3 b 0, w gt subdtrmiats of ar
İ t al: adamard Exotial al Matrix 87 from formula (8) Th, w gt Thus dt I B ( ) 0 0, ad roof is comltd Rfrcs [] R Rams, adamard ivrss, squar roots ad roducts of almost smidfiit matrics, Liar lgbra ad its licatios, 88 (999), 35-43 [] S Sola, D Bozurt, O th sctral orms of Cauchy-Tolitz ad Cauchy-al matrics, lid Mathmatics ad Comutatio, 40 (003), 3-38 [3] S Sola, D Bozurt, ot o boud for orms of Cauchy-al matrics, Numrical Liar lgbra with licatios, 0 (003), 377-38 [4] D Gügör, Lowr bouds for th orms of Cauchy-Tolitz ad Cauchy-al matrics, lid Mathmatics ad Comutatio, 57 (004), 599-604 [5] R Türm, D Bozurt, O th bouds for th orms of Cauchy-Tolitz ad Cauchy-al matrics, lid Mathmatics ad Comutatio, 3 (00), 633-64 [6] Nallı, O th adamard xotial GCD matrics, Slcu Joural of lid Mathmatics, 7() (006), 63-68 [7] G Zil, Som rmars o matrix orms, coditio umbrs ad rror stimats for liar quatios, Liar lgbra ad Its licatios, 0 (988), 9-4 [8] R Mathias, Th sctral orm of ogativ matrix, Liar lgbra ad Its licatios, 3 (990), 69-84