On the Explicit Determinants and Singularities of r-circulant and Left r-circulant Matrices with Some Famous Numbers

Transcription

1 O the Explicit Detemiats Sigulaities of -ciculat Left -ciculat Matices with Some Famous Numbes ZHAOLIN JIANG Depatmet of Mathematics Liyi Uivesity Shuaglig Road Liyi city CHINA JUAN LI Depatmet of Mathematics Liyi Uivesity Shuaglig Road Liyi city CHINA NUO SHEN Depatmet of Mathematics Liyi Uivesity Shuaglig Road Liyi city CHINA Abstact: Let A be a -ciculat matix B be a left -ciculat matix whose fist ows ae P P P Q Q Q J J J j j j espectively P is the Pell umbe Q is the Pell- Lucas umbe J is the Jacobsthal umbe j is the Jacobsthal-Lucas umbe I this pape by usig the ivese factoizatio of polyomial of degee the explicit detemiats of A B whose fist ows ae P P P Q Q Q ae expessed by utilizig oly Pell umbes Pell-Lucas umbes the paamete the explicit detemiats of A B whose fist ows ae J J J j j j ae expessed by utilizig oly Jacobsthal umbes Jacobsthal-Lucas umbes the paamete The esults ot oly exted the oigial esults but also simple i foms Also the sigulaities of those matices ae discussed Futhemoe fou idetities of those famous umbes ae give Key Wods: -ciculat matix Left -ciculat matix Detemiat Sigulaity Pell umbes Pell-Lucas umbes Jacobsthal umbes Jacobsthal-Lucas umbes Itoductio The Pell Pell-Lucas sequeces [] the Jacobsthal Jacobsthal-Lucas sequeces [] ae defied by the followig ecuece elatios espectively: P + P + P P 0 0 P Q + Q + Q Q 0 Q J + J + J J 0 0 J j + j + j j 0 j The fist few values of the sequeces ae give by the followig table 0: P Q J j The sequeces {P } {Q } {J } {j } ae give by the Biet fomulae P α β Q α + β J α β α β j α + β α β ae the oots of the chaacteistic equatio x x 0 α β ae the oots of the equatio x x 0 Defiitio [3] A -ciculat matix M M deoted by Cic a a a is a matix of the fom M : a a a a a a a a a a a 3 a a a 3 a a Note that the -ciculat matix is a ciculat matix [4] whe is a uppe Toeplitz matix whe 0 is a sew-ciculat matix [5] whe Defiitio [6] A left -ciculat matix N M deoted by LCic a a a is a matix of the E-ISSN: Issue 3 Volume Mach 03

2 fom a a a 3 a a a 3 a a N : a 3 a a a a a a a Lemma 3 [6] If M Cic a a a the det M j a j ε j j a j ε j ε ae the oots of the equatio x 0 Lemma 4 [6] Let M Cic a a a The M is osigula if oly if fx 0 fx gx j a j x j gx x fo The -ciculat matix play a impotat ole i vaious applicatios [ ] Boma [] peseted a simple deivatio of the Mooe-Peose pseudoivese of a abitay squae -ciculat matix Recetly thee ae may iteests i popeties geealizatio of some special matices ivolvig famous umbes Djodjević [] peseted a systematic ivestigatio of the icompletes geealized Jacobsthal Jacobsthal-Lucas umbes Melham [] gave some fomulae ivolvig Fiboacci Pell umbes She discussed the bouds of the oms of -ciculat matix with some famous umbes i [3 3] The authos discussed some popeties of special matices ivolvig Fiboacci o Lucas umbes i [4 5 6] itoduced cetai of geealizatios i [7 8] Jaiswal evaluated some detemiats of ciculat whose elemets ae the geealized Fiboacci umbes [9] Lid peseted the detemiats of ciculat sew-ciculat ivolvig Fiboacci umbes i [0] Li gave the detemiat of the Fiboacci-Lucas quasi-cyclic matices [] I this pape by usig the ivese factoizatio of polyomial of degee the explicit detemiats of the -ciculat left -ciculat matix ivolvig Pell umbes Pell-Lucas umbes ae expessed by utilizig oly Pell umbes Pell-Lucas umbes the paamete the explicit detemiats of the -ciculat left -ciculat matix ivolvig Jacobsthal umbes Jacobsthal-Lucas umbes ae expessed by utilizig oly Jacobsthal umbes Jacobsthal-Lucas umbes the paamete Also the sigulaities of those matices ae discussed by judgig whethe two give polyomials copime Futhemoe fou idetities of those famous umbes ae give Detemiats sigulaities of -ciculat left -ciculat matix with Pell umbes I this sectio we fist give a fomula the give the explicit detemiats of Cic P P P LCic P P P the discuss the sigulaities of them Lemma 5 y ε z y z ε satisfies the equatio y z C Poof Whe z 0 the esult is obvious Whe z 0 we deduce that y ε z z y z ε Sice ε satisfies the equatio so we must have x x ε By usig the ivese factoizatio of polyomial we obtai [ y ε z z y ] y z z Theoem 6 Let A Cic P P P The det A P + + P + Q E-ISSN: Issue 3 Volume Mach 03

3 Futhemoe A is sigula if oly if P + ρω P 0 α ρω β ρω 0 ρ ω cos π π + i si Poof We give the explicit detemiat of the matix A fistly By Lemma 3 ε satisfies the equatio we ca get P + P ε + + P ε + α β ε + + α β ε α β [ α + α ε + + α ε α β ] β + β ε + + β ε α α α ε β β β ε P + ε P α ε β ε det A λ P + ε P α ε β ε Accodig to Lemma 5 we have det A P + P α β P + + P + Q Next we discuss the sigulaity of the matix A If 0 the all the eigevalues of the matix A ae A is osigula If 0 the the oots of polyomial gx x ae ρω So we have ρ ω cos π π + i si fρω P + P ρω + + P ρω + α β ρω + + α β ρω [ α β ] α ρω β ρω P + ρω P α ρω β ρω By Lemma 4 the matix A is osigula if oly if fρω 0 That is whe α ρω β ρω 0 A is osigula if oly if P + ρω P 0 fo ay C Whe α ρω β ρω 0 we have ρω α o ρω β If ρω α the the eigevalue of A is α β [ β α ] β α P α β α 0 fo α + β N + If ρω β the the eigevalue of A is α β α β α β P β 0 fo α + β N + So A is osigula fo α ρω β ρω 0 Thus the poof is completed Theoem 7 Let B LCic P P P The det B P + P + Q Futhemoe B is sigula if oly if ρ ρp + ω P 0 ρ α ω ρ β ω 0 ρ ω cos π π + i si E-ISSN: Issue 3 Volume Mach 03

4 Poof We give the explicit detemiat of the matix B fistly Whe 0 the matix B ca be witte as B P P P P P P P P P Γ A Γ A Hece we have P P P P P P P P P det B det Γ det A A is a -ciculat matix its detemiat ca be gotte fom Theoem 6 by eplacig with So P + +P det A + Q Whe 0 det Γ det B det A det Γ P + +P + Q P + P + Q det B P P + P + Q Next we discuss the sigulaity of the matix B If 0 the det B P 0 fo ay N + B is osigula If 0 A is sigula if oly if P + ω P 0 α ω β ω 0 by Theoem 6 That is ρ ρp + ω P 0 ρ α ω ρ β ω 0 Futhemoe the matix Γ is osigula with 0 This completes the poof 3 Detemiats sigulaities of -ciculat left -ciculat matix with Pell-Lucas umbes I this sectio we fist give the explicit detemiats of Cic Q Q Q LCic Q Q Q the discuss the sigulaities of them Theoem 8 Let A Cic Q Q Q The det A Q + Q + Q Futhemoe A is sigula if oly if Q + + Q ρω 0 α ρω β ρω 0 ρ ω cos π π + i si Poof We give the explicit detemiat of the matix A fistly By Lemma 3 we obtai Q + Q ε + + Q ε α + β + α + β ε + + α + β ε α + α ε + + α ε β + β ε + + β ε α α α ε + β β β ε Q + + Q ε α ε β ε + E-ISSN: Issue 3 Volume Mach 03

5 det A λ Accodig to Lemma 5 we have Q + + Q ε α ε β ε det A Q + Q α β Q + Q + Q Next we discuss the sigulaity of A If 0 the all the eigevalues of A ae A is osigula If 0 the the oots of polyomial gx x ae ρω Thus we have ρ ω cos π π + i si fρω Q + Q ρω + + Q ρω α + α ρω + + α ρω +β + β ρω + + β ρω α α α ρω + β β β ρω Q + + Q ρω α ρω β ρω By Lemma 4 the matix A is osigula if oly if fρω 0 That is whe α ρω β ρω 0 A is osigula if oly if fo ay C Whe Q + + Q ρω 0 α ρω β ρω 0 we have ρω α o ρω β If ρω α the the eigevalue of the matix A is β α β α 0 fo α + β If ρω β the the eigevalue of the matix A is α β 0 fo α + β So the matix A is osigula fo α ρω β ρω 0 Hece the poof is completed Theoem 9 Let B LCic Q Q Q The det B Q + Q + Q Futhemoe B is sigula if oly if ρ ρq + + Q ω 0 ρ α ω ρ β ω 0 ρ ω cos π π + i si Poof We give the explicit detemiat of the matix B fistly Whe 0 the matix B ca be witte as B Q Q Q Q Q Q Q Q Q Γ A Thus we have Q Q Q Q Q Q Q Q Q det B det Γ det A Γ A E-ISSN: Issue 3 Volume Mach 03

6 A is a -ciculat matix its detemiat ca be obtaied fom Theoem 8 by eplacig with So det A Whe 0 + Q Q + det Γ Q det B det A det Γ + Q Q + Q Q + Q + Q det B Q Q + Q + Q Next we discuss the sigulaity of B If 0 the det B Q 0 fo ay N + B is osigula If 0 A is sigula if oly if Q + + Q ω 0 α ω β ω 0 by Theoem 8 That is ρ ρq + + Q ω 0 ρ α ω ρ β ω 0 Futhemoe the matix Γ is osigula with 0 Thus the poof is completed 4 Detemiats sigulaities of -ciculat left -ciculat matix with Jacobsthal umbes I this sectio we fist give the explicit detemiats of Cic J J J LCic J J J the discuss the sigulaities of them Theoem 0 Let A Cic J J J The det A J + + J + j Futhemoe A is sigula if oly if J + ρω J 0 α ρω β ρω 0 ρ ω cos π π + i si Poof We give the explicit detemiat of the matix A fistly By Lemma 3 ε satisfies the equatio we ca get J + J ε + + J ε α β + α β ε + + α β ε α β α β α β [ α + α ε + + α ε α β ] β + β ε + + β α β ε α α α ε β β β ε J + ε J α ε β ε det A λ J + ε J α ε β ε Accodig to Lemma 5 we have det A J + J α β J + + J + j Next we discuss the sigulaity of the matix A E-ISSN: Issue 3 Volume Mach 03

7 If 0 the all the eigevalues of the matix A ae A is osigula If 0 the the oots of polyomial gx x ae ρω So we have ρ ω cos π π + i si fρω J + J ρω + + J ρω α β + α β ρω + + α β α β α β ρω α β [ α β ] α β α β α ρω β ρω J + ρω J α ρω β ρω By Lemma 4 the matix A is osigula if oly if fρω 0 That is whe α ρω β ρω 0 A is osigula if oly if fo ay C Whe J + ρω J 0 α ρω β ρω 0 we have ρω α o ρω β If ρω α the the eigevalue of A is [ ] β α β α β α J α β α 0 fo α β N + If ρω β the the eigevalue of the matix A is ] α β [ β α α β J β α β 0 fo α β N + So A is osigula fo α ρω β ρω 0 The poof is the completed Theoem Let B LCic J J J The det B J + J + j Futhemoe B is sigula if oly if ρ ρj + ω J 0 ρ α ω ρ β ω 0 ρ ω cos π π + i si Poof We give the explicit detemiat of the matix B fistly Whe 0 the matix B ca be witte as B J J J J J J J J J Γ A Γ A 3 Hece we have J J J J J J J J J det B det Γ det A 3 A 3 is a -ciculat matix its detemiat ca be gotte fom Theoem 0 by eplacig with J + +J det A 3 + j det Γ E-ISSN: Issue 3 Volume Mach 03

8 So Theoem Let A Cic j j j The Whe 0 det B det A 3 det Γ J + +J + j J + J + j det B J J + J + j Next we discuss the sigulaity of the matix B If 0 the det B J 0 fo ay N + B is osigula If 0 A 3 is sigula if oly if J + ω J 0 α ω β ω 0 by Theoem 0 That is ρ ρj + ω J 0 ρ α ω ρ β ω 0 Futhemoe the matix Γ is osigula with 0 This completes the poof 5 Detemiats sigulaities of -ciculat left -ciculat matix with Jacobsthal-Lucas umbes I this sectio we fist give the explicit detemiats of Cic j j j LCic j j j the discuss the sigulaities of them det A j + j + j Futhemoe A is sigula if oly if j + + j ρω 0 α ρω β ρω 0 ρ ω cos π π + i si Poof We give the explicit detemiat of the matix A fistly By Lemma 3 we obtai j + j ε + + j ε α + β + α + β ε + + α + β ε α + α ε + + α ε β + β ε + + β ε α α α ε + β β β ε j + + j ε α ε β ε det A λ + j + + j ε α ε β ε Accodig to Lemma 5 we have det A j + j α β j + j + j Next we discuss the sigulaity of A If 0 the all the eigevalues of A ae A is osigula If 0 the the oots of polyomial gx x ae ρω Thus we have ρ ω cos π π + i si fρω j + j ρω + + j ρω E-ISSN: Issue 3 Volume Mach 03

9 α + α ρω + + α ρω +β + β ρω + + β ρω α α α ρω + β β β ρω j + + j ρω α ρω β ρω By Lemma 4 the matix A is osigula if oly if fρω 0 That is whe α ρω β ρω 0 A is osigula if oly if j + + j ρω 0 fo ay C Whe α ρω β ρω 0 we have ρω α o ρω β If ρω α the the eigevalue of the matix A is β α β α 0 fo α β If ρω β the the eigevalue of the matix A is α β α β 0 fo α β So the matix A is osigula fo α ρω β ρω 0 Hece the poof is completed Theoem 3 Let B LCic j j j The det B j + j + j Futhemoe B is sigula if oly if ρ ρj + + j ω 0 ρ α ω ρ β ω 0 ρ ω cos π π + i si Poof We give the explicit detemiat of the matix B fistly Whe 0 the matix B ca be witte as B j j j j j j j j j Γ A Γ A 4 Thus we have j j j j j j j j j det B det Γ det A 4 A 4 is a -ciculat matix its detemiat ca be obtaied fom Theoem by eplacig with So det A 4 + j j + det Γ j det B det Γ det A 4 j + j + j Whe 0 det B j j + j + j Next we discuss the sigulaity of B If 0 the det B j 0 fo ay N + B is osigula If 0 A 4 is sigula if oly if j + + j ω 0 E-ISSN: Issue 3 Volume Mach 03

10 α ω β ω 0 by Theoem That is ρ ρj + + j ω 0 ρ α ω ρ β ω 0 Futhemoe the matix Γ is osigula with 0 Thus the poof is completed 6 Coclusio I this sectio we give two idetities of Pell Pell- Lucas umbes two idetities of Jacobsthal Jacobsthal-Lucas umbes Let C CicP P P D CicQ Q Q be ciculat matices Jiag [] got detc P + +P P+ P P det D [ Q + + Q ] Q + Q + 3Q + Q + We have det C P + P + Q i Theoem 6 det D Q + Q + Q i Theoem 8 whe Similaly let E CicJ J J F Cicj j j be ciculat matices Gog [3] got det E J + + J J+ J J det F j + + j 4 3 j+ 3 j + 5j + j 4 We have det E J + J + j i Theoem 0 det F j + j + j i Theoem whe So we have the followig idetities of P Q of J j : P + + P P+ P P P + P + Q 3 ] [ Q + + Q Q + Q + Q 4 J + + J J + J + j 5 j + + j 4 3 j + j + j 6 Q + Q + 3Q + Q + J+ J J j+ 3 j + 5j + j 4 Acowledgemets: This poject is suppoted by the Pomotive Reseach Fud fo Excellet Youg Middle-aged Scietists of Shog Povice Gat No BS 0DX004 E-ISSN: Issue 3 Volume Mach 03

11 Refeeces: [] R Melham Sums ivolvig Fiboacci Pell umbes Pot Math pp [] T Hozum O some popeties of Hoadam polyomials It Math Foum pp 43 5 [3] S She J Ce O the bouds fo the oms of -ciculat matices with the Fiboacci Lucas umbes Appl Math Comput 6 00 pp [4] S She J Ce Y Hao O the detemiats iveses of ciculat matices with Fiboacci Lucas umbes Appl Math Comput 7 0 pp [5] J Lyess T Söevi Fou-dimesioal lattice ules geeated by sew-ciculat matices Math Comput pp [6] Z Jiag Nosigulaity o two sots of ciculat matices Math Pactice Theoy 995 pp 5 58 [7] S Noschese L Reichel Geealized ciculat Stag-type pecoditioes Nume Liea Algeba Appl 9 0 pp 3 7 [8] I Hwag D Kag W Lee Hypoomal Toeplitz opeatos with matix-valued ciculat symbols Complex Aal Ope Theoy DOI: 0007/s [9] D Faeic W Lee O hypoomal Toeplitz opeatos with polyomial ciculat-type symbols Iteg Equ Ope Theoy pp 0 0 [0] D Betaccii M K Ng Bloc {ω}- ciculat pecoditioes fo the systems of diffeetial equatios Calcolo pp 7 90 [] E Boma The Mooe-Peose pseudoivese of a abitay squae -ciculat matix Liea Multiliea Algeba pp [] G B Djodjević H M Sivastava Icomplete geealized Jacobsthal Jacobsthal- Lucas umbes Math Comput Model pp [3] S She J Ce O the spectal oms of - Ciculat matices with the -Fiboacci - Lucas Numbes It J Cotemp Math Scieces 5 00 pp [4] G Lee J Kim S Lee Factoizatios eigevalues of Fiboacci symmetic Fiboacci matices Fiboacci Quat pp 03 [5] Z Zhag Y Zhag The Lucas matix some combiatoial idetities Idia J Pue Appl Math pp [6] M Abula D Bozut O the oms of Toeplitz matices ivolvig Fiboacci Lucas umbes Hacet J Math Stat pp [7] P Staimiović J Niolov I Staimioviá A geealizatio of Fiboacci Lucas matices Discete Appl Math pp [8] M Miladiović P Staimiović Sigula case of geealized Fiboacci Lucas matices J Koea Math Soc 48 0 pp [9] D Jaiswal O detemiats ivolvig geealized Fiboacci umbes Fiboacci Quat pp [0] D Lid A Fiboacci ciculat Fiboacci Quat pp [] D Li Fiboacci-Lucas quasi-cyclic matices Fiboacci Quat pp [] Z Jiag Y Gog Y Gao O the detemiats ivese of ciculat matices with the Pell Pell-Lucas umbes submited to Appl Math Comput [3] Y Gog Z Jiag O the detemiats iveses of ciculat matices with the Jacobsthal Jacobsthal-Lucas umbes submited to WSEAS Tas Math E-ISSN: Issue 3 Volume Mach 03