Some Results on Certain Symmetric Circulant Matrices

Transcription

1 Joural of Iformatics ad Mathematical Scieces Vol 7, No, pp 81 86, 015 ISSN olie; X prit Pulished y RGN Pulicatios Some Results o Certai Symmetric Circulat Matrices Research Article AV Ramakrisha 1, * ad TVN Prasaa 1 Departmet of Mathematics, RVR ad JC College of Egieerig, Chowdavaram, Gutur 5019, Adhra Pradesh, Idia Departmet of Mathematics, NRI Istitute of Techology, Visadala, Gutur 5009, Adhra Pradesh, Idia *Correspodig author: amathi7@gmailcom Astract A direct method for fidig the iverse of a class of symmetric circulat matrices is give i [4] I this paper a method of fidig the Moore-Perose iverse for a class of sigular circulat matrices is preseted ad the spectral orm ad spectral radius are calculated Fially the spectral orm ad spectral radius for symmetric circulat matrices with iomial coefficiets are derived Keywords Symmetric circulat matrix; Moore-Perose iverse; Spectral radius MSC 05A19; 15A15 Received: March 3, 015 Accepted: Jue 19, 015 Copyright 015 AV Ramakrisha ad TVN Prasaa This is a ope access article distriuted uder the Creative Commos Attriutio Licese, which permits urestricted use, distriutio, ad reproductio i ay medium, provided the origial work is properly cited 1 Itroductio I [4] the authors studied some properties icludig iversio ad applicatios of a class of symmetric circulat matrices Whe the matrix is sigular, the Moore-Perose iverse replaces the iverse I Sectio we calculate the Moore-Perose iverse for matrices a The cocept of parallel sum of two hpsd see ref 3 matrices was itroduced y WN Aderso, Jr ad RJ Duffi [1] The parallel sum of matrices is useful i electrical etworks We calculate the parallel sum of two hpsd matrices i this sectio I Sectio 3 the spectral orm ad spectral radius for symmetric circulat matrices with iomial coefficiets are derived usig some Baach algera results

2 8 Some Results O Certai Symmetric Circulat Matrices: AV Ramakrisha ad TVN Prasaa A circulat matrix A = a 1 a a 3 a of order is symmetric if i a + j = a j, j, whe is eve, ii a + j+ 1 = a + 3 j 1, 1 j, whe is odd I particular a matrix A = a of order ad the matrix C = C0 C1 C C 1 with Biomial coefficiets are symmetric circulat [ ] We have [1],, ,, are symmetric circulat matrices with iomial coefficiets Give a matrix A with real or complex etries there exists a uique matrix X satisfyig the Moore-Perose equatios: 1 AX A = A, X AX = X, 3 AX = AX, 4 X A = X A, where deotes the cojugate traspose Such a uique X correspodig to A is called the Moore-Perose iverse of A ad is deoted y A Moore-Perose Iverse of Certai Sigular Symmetric Circulat Matrix Theorem 1 For two distict real umers a, with a + 1 = 0, the Moore-Perose A of A = a of order is A 1 = a a Proof Sice a + 1 = 0, clearly A is sigular, we ca fid the Moore-Perose iverse of A as follows: Whe A = a, A = c d d d, where c ad d are give y c = a + 1, d = a + If X = x 1 x x x, sice A, X are circulat, AX = X A Hece AX A = A X = c d d d x 1 x x x = cx 1 + 1dx dx 1 + c + 1dx dx 1 + c + 1dx Joural of Iformatics ad Mathematical Scieces, Vol 7, No, pp 81 86, 015

3 Some Results O Certai Symmetric Circulat Matrices: AV Ramakrisha ad TVN Prasaa 83 Hece AX A = A = cx 1 + 1dx dx 1 + c + 1dx dx 1 + c + 1dx = a cx 1 + 1dx = a, dx 1 + c + 1dx = a x 1 a x = 1 x 1 x = 1 a Thus Whe so that where AX A = A = x 1 x = 1, sice a + 1 = 0 a x 1 = x + 1 a, X = x + a x 1 + a X A = x + a x 1 + a = c d d d c = ax + d = x + = a x + x, a x a + x a a + ax a + 1x + 1x, a a + x a + ax a + ax + x a a a a Hece X AX = X iff a a a Comparig Thus x 1 = a a a, x = a a X = a a a = x 1 x x x a Sice A ad X are symmetric circulat matrices, X A = X A = AX Hece the Moore-Perose iverse A 1 = a a + x Joural of Iformatics ad Mathematical Scieces, Vol 7, No, pp 81 86, 015

4 84 Some Results O Certai Symmetric Circulat Matrices: AV Ramakrisha ad TVN Prasaa Corollary The Moore-Perose iverse of a symmetric circulat matrix is symmetric circulat Defiitio 3 A square matrix A is said to e Hermitia positive semi-defiitehpsd if A = A ad all the eigevalues of A are greater tha or equal to zero ad at least oe eigevalue is zero Lemma 4 If A = a 1 a a a, B = 1 are two symmetric circulat matrices of order with a 1 > a, a 1 + 1a = 0 ad 1 >, = 0 the A + B is hpsd Proof Sice a 1 > a, a 1 + 1a = 0, we have all the eigevalues of A are greater tha or equal to 0, so that A is hpsd Similarly B is also hpsd Now A + B = a a + a +, clearly A + B is also hpsd Defiitio 5 The parallel sum of two matrices A ad B is defied as AA + B B Theorem 6 If A = a 1 a a a, B = 1 satisfyig the coditios 1 of Lemma 4 the the parallel sum of A ad B is AA + B B a 1 a + 1 Proof Sice A ad B are hpsd, A + B is also hpsd Now A + B = a a + a + By Theorem 1, A + B 1 = a1 a 1 a a + a + 1 Therefore the parallel sum is AA + BB a 1 a + 1 Theorem 7 For two distict elemets a, i the complex field, the spectral orm of the symmetric circulat matrix A = a of order is ρa = max{ a+ 1, a } Proof From [] the eigevalues of A are a + 1, a Thus the spectrum of A is σa = {a + 1, a } Hece the spectral orm of A is ρa = max{ a + 1, a } More over if A is sigular the a + 1 = 0 ad hece spectral radius of A is a Remark 8 Perro ad Froeius theorem [3] states that if A = a i j is a matrix with oegative elemets a i j 0, there exists a eigevector of A with o-egative coordiates ad with eigevalue ρ, such that all other eigevalues satisfy λ ρ I particular if A = a of order ad if a, are oegative the a + 1 is the maximum eigevalue of A ad y Perro ad Froeius theorem a eigevector correspodig to a + 1 cotais o-egative coordiates We ca choose this eigevector as 1,1,1,,1 Theorem 9 [] For ay elemet of a Baach Algera X of the form e z with z < 1 there exists a uique y = e x i X with x < 1 such that y = e z ie, e x = e z I other words e z has a square root Joural of Iformatics ad Mathematical Scieces, Vol 7, No, pp 81 86, 015

5 Some Results O Certai Symmetric Circulat Matrices: AV Ramakrisha ad TVN Prasaa 85 This could e used to estalish the followig theorem: Theorem 10 For two distict elemets a, i a Baach algera X of all complex matrices with a < 1 ad < 1 the of symmetric circulat matrix I A, where A = a of order has a square root I B i X Proof Sice the eigevalues of A = a are λ = a + 1, a, a,, a 1 times For a < 1 ad < 1, we have each eige value λ of A satisfies λ < 1 Thus the spectral radius ρa = max { λ i : i = 1,, } is less tha 1 Therefore y Theorem 9 there exists a uique I B such that I B = I A i X Hece I B is the square root of I A 3 Symmetric Circulat Matrices with Biomial Coefficiets Theorem 31 The eigevalues of the symmetric circulat matrix C = C0 C1 C C 1 are give y 1, cos πm 1, m = 1,,3,, 1 Proof We have the eigevalues of a circulat matrix A = a 0, a 1, a,, a 1 are ie, 1 πik λ j = a j e j, k = 0,1,,, 1 1 λ j = a j cos πk j Sice C is symmetric, 1 + i a j si πk j 1 λ k = C j cos πk j, k = 0,1,,, 1 Thus the eige values of C 1 are Also detc = 1 λ 0 = C0 + C1 + C + + C 1 = 1,, k = 0,1,,, 1 λ 1 = C0 + C1 cos π + C cos 4π + + 1π C 1 cos λ = C0 + C1 cos 4π + C cos 8π + C 3 cos 1π + = 1π λ = 1 + cos 1 cos πm 1 m=0 = 1 + cos π 1, 1, 1 + cos 4π Joural of Iformatics ad Mathematical Scieces, Vol 7, No, pp 81 86, 015

6 86 Some Results O Certai Symmetric Circulat Matrices: AV Ramakrisha ad TVN Prasaa Theorem 3 The spectral orm of C is ρc = max{ 1, cos π 1 } Proof From Theorem 31 we have its spectrum is Now Sice cos πm { σc = 1, 1 + cos π 1, 1 + cos 4π 1π 1,, 1 + cos 1} { = 1, cos π, cos π } 1,, 1π cos 1 { ρc = max 1, cos π, 1 cos π { is maximum for m = 1 we get ρc = max 1,, 1π } cos 1 1, cos π } 1 Remark 33 I C all the elemets are positive ad 1 is the maximum eigevalue So y Perro ad Froeius theorem a eigevector correspodig to 1 cotais o-egative coordiates We ca choose this eigevector as 1,1,1,,1 Ackowledgemet The authors are thakful to Prof I Ramahadra Sarma ad Dr D V Lakshmi for their valuale commets ad suggestios Competig Iterests The authors declare that they have o competig iterests Authors Cotriutios Both the authors cotriuted equally ad sigificatly i writig this article as well as read ad approved the fial mauscript Refereces [1] WN Aderso, Jr ad RJ Duffi, Series ad parallel additio of matrices, Joural of Mathematical Aalysis ad Applicatios , [] FF Bosall ad DSG Stirlig, Square roots i Baach -algeras, Glasgow Math J 13 74, 197 [3] RA Hor ad CR Johso, Matrix Aalysis, Camridge Uiversity Press 1985 [4] AV Ramakrisha ad TVN Prasaa, Symmetric circulat matrices ad pulickey cryptography, It J Cotemp Math Scieces , Joural of Iformatics ad Mathematical Scieces, Vol 7, No, pp 81 86, 015