It J Cotep Math Scieces Vol 5 o 5 3 - O the Ciculat Matices with Aithetic Sequece Mustafa Bahsi ad Süleya Solak * Depatet of Matheatics Educatio Selçuk Uivesity Mea Yeiyol 499 Koya-Tukey Ftly we have defied Ca ( cij) Abstact as x atix whee a + ( j i od ) a ad ae eal ubes The we have studied the eigevalues deteiat spectal o Euclidea o of the atix C Also we have ivestigated the spectal o Euclidea o of ivese of the atix C c ij Matheatics Subject Classificatio: 5B5 5A5 5A6 Keywods: Ciculat Matix Eigevalue No Deteiat Itoductio A x atix C called a ciculat atix if it of the fo * coespodig autho: ssolak@selcukedut
4 M Bahsi ad S Solak c c c L c c c c c c 3 c L C M M M L M M c c3 c4 L c c c c c3 L c c Fo each i j K ad i j such that j i k( od ) have the sae value c k ; these eleets fo the so-called k th stipe of C Obviously a ciculat atix deteied by its ft ow ( o colu ) That C ci c c K c ) ( k K all the eleets ( ) The popeties of ciculat atices ae well kow[63] Let w be a piiitive th oot of the uity ad ( ) f k fo the sequece ( k ) k λ c k λ be the geeatig polyoial c ( the zeoth ow of the atix C ) The fo evey K f ( w ) a eigevalue of the atix C ad x ( ) ( w w w ) T K a eigevecto of C belogig to the eigevalue f ( w ) equivalet to the diagoal atix with diagoal eties ( w ) K [5] Also the atix C oal ad det C f ( w ) [3] These eigevectos ae othogoal ad thus the atix C uitaily πi f Let w deote w e The w a piitive -th oot of uity πi Theefoe fo evey K πik λ f e cke a k πi πi πi( ) T eigevalue of the atix C ad x e e e K a eigevecto of C belogig to the eigevalue λ May authos have bee studyig ciculat atices Hladik[5] gave a foule fo Schu o of a block ciculat atix with ciculat blocks Kae at al[4] woked o spectal decopositios ad sigula value decopositios of fou types of eal ciculat atices Bose ad Mita[] deived the liitig spectal dtibutio of a paticula vaiat of a ciculat ado atix Atki at al[] studied the powes of a ciculat Zhag at al[7] woked o the iial polyoials ad iveses of a block ciculat atices ove a field Solak [8]
O the ciculat atices with aithetic sequece 5 deived soe bouds fo the spectal o ad Euclidea o of ciculat atices with the Fiboacci ad Lucas ubes I th pape we have defied Ca cij such that c ij a + ( j i od ) a ad ae eal ubes The we have studied the eigevalues deteiat spectal o Euclidea o of the atix C ad ivese of th atix ciculat atices ( ) Mai Results Theoe The eigevalues of the x atix C ae whee ( ) λ a + λ i π e Poof Sice C a ciculat atix its eigevalues ae of the fo k ( a + k) πik λ e fo evey Fo we have Let x be λ k a + ax πi k ( a + k) λ a + a + + a + + L + a + ( ) ( ) a + x e Fo we have πik ( a + k) e ( a + k) k x k 3 ( + x + x + L + x ) + x( + x + 3x + 4x + L+ ( ) x ) x x d a + a x x + x d x x a + a + x x dx x 3 4 ( x + x + x + x + L+ x ) dx
6 M Bahsi ad S Solak a a + x ( x )( x) + ( x x ) ( ) x x + x + x ( x ) ( ) ( ) x x ( x) π i ( x ) e Theoe The spectal o of the x atix C ( ) Ca ax a+ π si Poof Sice a ciculat atix a oal atix we ca wite C ( ) ( ) ax λ + ( ) ax a ax λ : : Thus fo evey λ e πi π π cos + i si π cos π cos π si π cos + + si π π π 4 si si Fo λ has axiu value Theefoe
O the ciculat atices with aithetic sequece 7 ad ax : ( λ ) π si ( ) Ca ax a+ π si Theoe 3 The Euclidea o of the x atix C ( ) ( )( ) Ca a a 6 + + E Poof Fo the defiitio of the Euclidea o Thus ( ) E + + s s C a s a s + + a a s s s s s ( ) ( ) ( ) a + a + 6 ( ) ( )( ) a + a + 6 ( ) ( )( ) Ca a a 6 + + E Theoe 4 The deteiat of the x atix C ( ) Ca a + Poof If we apply the popeties of the deteiat to the deteiat of the atix C the we have the followig equalities:
8 M Bahsi ad S Solak C a a+ a+ a+ 3 L a+ ( 3) a+ ( ) a+ ( ) ( ) ( ) ( ) L L M M M M L M M M 3 3 3 3 L (3 ) (3 ) (3 ) L ( ) ( ) L ( ) a 3 L ( 3) ( ) ( ) L L M M M M L M M M L L L If we calculate the deteiat of the above atix we have ( ) L Ca a ( ) ( ) ( ) ( ) ( ) [ ( a + + + 3 + L + ( ) )] ( ) ( ) a + ( ) ( ) a + ( ) Theoe 5 The adjoit of the x atix C 3 3 + 3 a+ a ci 3 ( ) ( ) Adj C
O the ciculat atices with aithetic sequece 9 Poof Sice the adjoit of a ciculat atix also ciculat we will calculate cofactos of the eleets of the ft colu of the atix C Theefoe we will obtai the eleets of the ft ow of the atix Adj ( C ) c ( j K ) If we calculate the cofactos of the eleets j atix C we have espectively of the 3 3 + ( ) a + ( ) a 3 3 3 ( ) ( ) ( ) ( ) ( ) ( ) Theefoe 3 3 3 ( C ) ( ) + a+ a Adj ci 3 Theoe 6 The ivese of the x atix C + a + a + C ci a + Poof Sice tivial C ( ) adj C fo theoe 4 ad theoe 5 the poof det C Coollay The eigevalues of the x atix equalities ψ ( ) a + whee Theoe 7 The spectal o of the x atix C C πi e ψ C π si ax ( ) a + ae satfy the followig
M Bahsi ad S Solak whee deotes exact value Poof Sice the ivese of a ciculat atix also a oal atix we ca wite C ( ) ax ψ ( ) ax ax ψ : ( ) : a + Fo evey ψ πi e π π cos + i π cos π cos π π cos + + si Sice the values π si π 4si π si π si ad ψ ae axiu fo Theefoe ad ax : ( ψ ) π si
O the ciculat atices with aithetic sequece π si Ca ax ( ) a + Theoe 8 The Euclidea o of the x atix C C a + E a + Poof Fo the defie of Euclidea o we have + a + a + + Ca E + + + L + 4 a + 4 3 4 + 3 a + + a a 4 + 3 a + + a a a + + + 3 + 4 a a a + + 4 + a + Thus the poof copleted Refeeces [] AOL Atki ad E Boos ad K Cechlaova U N Peled Powes of Ciculats i Bottleack Algeba Liea Algeba ad Its Applicatios 58
M Bahsi ad S Solak (997)37-48 [] Aup Bose ad Joydip Mita Liitig Spectal Dtibutio of a Special Ciculat Stattics & Pobability Lettes 6 () - [3] FZhag Matix Theoy (Basic Results ad Teciques) Spige-Velag999 [4] H Kae ad J Scheid ad C W Uebehube Spectal Decopositio of Real Ciculat Matices Liea Algeba ad Its Applicatios 367(3) 3-3 [5] M Hladik Schu Nos of Biciculat Matices Liea Algeba ad Its Applicatios 86 (999) 6-7 [6] P J Dav Ciculat Matices Wiley 979 [7] SZhag ad ZJiag ad S Liu A Applicatio of the Göbe Bas i Coputatio fo the Miial Polyoials ad Iveses of Block Ciculat Matices Liea Algeba ad Its Applicatios 347() -4 [8] SSolak O the Nos of Ciculat Matices with the Fiboacci ad Lucas Nubes Applied Matheatics ad Coputatios 6(5) 5-3 Received: Novebe 9