Right circulant matrices with ratio of the elements of Fibonacci and geometric sequence

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1 Notes o Number Theory ad Discrete Mathematics Prit ISSN , Olie ISSN Vol. 22, 2016, No. 3, Right circulat matrices with ratio of the elemets of Fiboacci ad geometric sequece Aldous Cesar F. Bueo Philippie Sciece High School-Cetral Luzo Campus Lily Hill, Clark Freeport Zoe, Pampaga, Philippies afbueo@clc.pshs.edu.ph Received: 29 September 2015 Revised: 1 October 2016 Accepted: 2 October 2016 Abstract: We itroduce the right circulat matrices with ratio of the elemets of Fiboacci ad geometric sequece. Furthermore, we ivestigate their eigevalues, determiat, Euclidea orm, ad iverse. Keywords: Right circulat matrix with ratio of the elemets of Fiboacci ad geometric sequece, Eigevalues, Determiat, Euclidea orm, Matrix iverse. AMS Classificatio: 15B05. 1 Itroductio It has bee a traditio to use umber sequeces as etries for right circulat matrices. Works o these special matrices ormally would cocetrate o the ivestigatio of their eigevalues, determiats, Euclidea orms, spectral orms ad iverses. Recet works o this study iclude that of Bahsi ad Solak?] ad Bueo?],?]. Bahsi ad Solak used arithmetic sequece as etries. They provided the explicit formulas for the determiat, eigevalues, Euclidea ad spectral orms, ad iverses of these special type of right circulat matrices. They also ivestigated the Euclidea ad spectral orms of the iverses of these matrices. Bueo also did the same i?] (usig geometric sequece) ad i?] (usig Fiboacci sequece). I this study, we will use the sequece {s k } + k0 where s k F k is the ratio of the kth Fiboacci ar k umber ad the kth elemet of geometric sequece. Hece, the right circulat matrix takes the 79

2 form C R ( s) s 0 s 1 s 2... s 2 s 1 s 1 s 0 s 1... s 3 s 2 s 2 s 1 s 0... s 4 s s 2 s 3 s 4... s 0 s 1 s 1 s 2 s 3... s 1 s 0. As a cotiuig traditio o the study of right circulat matrices, we will derive the explicit forms of the eigevalues, determiat, Euclidea orm ad iverse of the matrix C R ( s). For the rest of the paper we will use C R ( s), C R ( s) E ad C 1 R ( s) to deote the determiat, the Euclidea orm ad the iverse of C R ( s). 2 Mai results We ow preset the mai results of our study. Theorem 2.1. The eigevalues of C R ( s) are give by λ m rf (F 1 r )ω m ar 1 (r φω m )(r ψω m ) (1) where m 0, 1,..., 1, φ , ψ ad ω e 2πi/. Proof: The discrete Fourier trasform is used to solve the eigevalues of right circulat matrices. Solvig for the eigevalues of C R ( s) yields λ m 1 k0 1 F k ar k ω mk φ k ψ k k0 ar k ω mk where m 0, 1,..., 1, φ 1+ 5, ψ 1 5 ad ω e 2πi/. 2 2 Cotiuig we have 1 1 ( ) φω m k ( ) ] ψω m k λ m a 5 r r k0 1 1 (φ/r) a 5 1 φω m /r 1 ] (ψ/r) 1 ψω m /r ] 1 r φ ar 1 5 r φω r ψ m r ψω m r(φ ψ ) + r ω m (φ ψ) ω m (φ 1 ψ 1 ) ar 1 5(r φω m )(r ψω m ) 80

3 Theorem 2.2. where L φ + ψ, the th Lucas umber. rf + r F 1 ω m F 1 ω m ar 1 5(r φω m )(r ψω m ) rf (F 1 r )ω m ar 1 (r φω m )(r ψω m ). C R ( s) ( 1) r F (r F 1 ) a r ( 1) (r 2 L + ( 1) ) (2) Proof: Recall that the determiat is the product of the eigevaluesm so we have Note that for ay x ad y, C R ( s)) 1 m0 rf (F 1 r )ω m ar 1 (r φω m )(r ψω m ) 1 (x yω m ) x y m0 Usig this relatioship results to Theorem 2.3. where A 2r2+2 r 2 L 2 +r 2 L 2 +L 2 2 r 4 r 2 L 2 +1 Proof: C R ( s) E ( 1) r F (r F 1 ) C R ( s) a r ( 1) (r 2 r (φ + ψ ) + ( 1) ) ( 1) r F (r F 1 ) a r ( 1) (r 2 L + ( 1) ) C R ( s) E 5 A 2B ] ar 1 ad B r2 ( 1) r φ k ψ k 2 ar k 5 k0 1 ] φ 2k + ψ 2k 2( 1) k 5a 2 r 2k m0 1 φ 2 /r ψ2 /r 2 2(1 ] ( r 2 )) 5a 2 1 φ 2 /r 2 1 ψ 2 /r r 2 ( )] r 2 φ 2 r2 ψ 2 r 2 ( 1) 2 5a 2 r 2( 1) r 2 φ 2 r 2 ψ 2 r (3) 81

4 Simplifyig further, we obtai C R ( s) E 5 2r 2+2 r 2 L 2 +r 2 L 2 +L 2 2 r 4 r 2 L ar 1 ( r 2 ( 1) r 2 +1 )] Theorem 2.4. where C 1 R C R (t 0, t 1,..., t 1 ) (4) t k ar 1 1 m0 (r φω m )(r ψω ) rf + (F 1 r )ω m (5) where k 0, 1,..., 1 ad provided that (r F 1 )ω m rf. Proof: The first row etries of the iverse of a right circulat matrix is just the iverse discrete Fourier trasform of the reciprocal of the eigevalues. Hece, we will have t k 1 1 m0 λ 1 m ω mk where k 0, 1,..., 1. Note that for the iverse to exist, all eigevalues should be o-zero. So for all m, (r F 1 )ω m rf. The above equatio results to t k ar 1 1 m0 (r φω m )(r ψω ) rf + (F 1 r )ω m 3 Coclusio I summary, we have obtaied the closed form of the eigevalues, determiat, Euclidea orm ad the iverse of C R ( s). Furthermore, we have the followig observatios: The eigevalues, the determiat ad the iverse of C R ( s) are all fuctios of F, F 1, φ, ψ, a, r ad. The Euclidea orm of C R ( s) is a fuctio of L 2, L 2, L 2 2, a, r ad. 82

5 Refereces 1] Bueo, A. (2012) Right Circulat Matrices with Geometric Progressio, Iteratioal Joural of Applied Mathematical Research, 1(4), ] Bueo, A. (2012) Right Circulat Matrices with Fiboacci Sequece, Iteratioal Joural of Mathematics ad Scietific Computig, 2(2), ] Bahsi, M., & Solak, S. (2010) O the Circulat Matrices with Arithmetic Sequece, Iteratioal Joural of Cotemporary Mathematical Scieces, 5(25),