Hidawi Publishig orporatio ISRN omputatioal Mathematics, Article ID 4570, 5 pages http://dx.doi.org/0.55/04/4570 Research Article Powers of omplex Persymmetric Atitridiagoal Matrices with ostat Atidiagoals Haibo Wag ollege of Sciece, Uiversity of Shaghai for Sciece ad Techology, Shaghai 00093, hia orrespodece should be addressed to Haibo Wag; uywaghaibo@6.com Received 3 Jauary 04; Accepted 3 March 04; Published 6 March 04 Academic Editors: F. W. S. Lima ad Q.-W. Wag opyright 04 Haibo Wag. This is a ope access article distributed uder the reative ommos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. We derive a geeral expressio for the pth power (p N of ay complex persymmetric atitridiagoal Hakel (costat atidiagoals matrices. Numerical examples are preseted, which show that our results geeralize the results i the related literature (Rimas 008, Wu 00, ad Rimas 009.. Itroductio Solvig some differece, differetial, ad delay differetial equatios, we meet the ecessity to compute the arbitrary positiveitegerpowersofsquarematrix.recetly,computig the iteger powers of atitridiagoal matrices has bee a very popular problem. There have bee several papers o computig the positive iteger powers of various kids of square matrices by Rimas et al., ad others [ 5]. I 0, the geeral expressio for the etries of the power of complex persymmetric or skew-persymmetric atitridiagoal matrices with costat atidiagoals is preseted by Gutiérrez- Gutiérrez []. Rimas [] gave the geeral expressio of the pth power for this type of symmetric odd order atitridiagoal matrices (atitridiag (, 0, i008.i[3, 4] asimilar problem is solved for atitridiagoal matrices havig zeros i mai skew diagoal ad uits i the eighbourig diagoals. I 00, the geeral expressio for the etries of the power of odd order atitridiagoal matrices with zeros i mai skew diagoal ad elemets,,...,;,,..., i eighbourig diagoals is derived by Rimas [5]. I 03, Rimas [6] gave the eigevalue decompositio for real odd order skew-persymmetric atitridiagoal matrices with costat atidiagoals (atitridiag (a,b,a ad derived the geeral expressio for iteger powers of such matrices. I the preset paper, we derive a geeral expressio for the pth power (p N of ay complex persymmetric atitridiagoal matrices with costat atidiagoals (atitridiag (b, a, b.thisovelexpressioisboth a extesio of the oe obtaied by Rimas for the powers of the matrix atitridiag (, 0, with N (see [] for theoddcasead[5] for the eve case ad a extesio of the oe obtaied by Hogli Wu for the powers of the matrix atitridiag (,, with N(see [3] fortheeve case.. Derivatio of Geeral Expressio I this preset paper, we study the etries of positive iteger power of a complex persymmetric atitridiagoal matrix with costat atidiagoals as follows: b a B atitridiag (b, a, b ( c c c, ( a b ( where a, b \ (0.
ISRN omputatioal Mathematics osider the followig complex Toeplitz tridiagoal matrix: a b A tridiag (b, a, b (. ( d d d ( b a The ext trivial result relates the matrix B with A ad with the backward idetity []: J (δ j, k+ j, ( where δ is the Kroecker delta. ( Lemma. Let a, b \ (0,ad N.The c, (3 B J A, (4 where B atitridiag (b, a, b ad A tridiag (b, a, b. Proof. We have [J A ] j,k h [J ] j,h [A ] h,k [A ] + j,k b, if + (j+k, a, if + (j+k0, b, if + (j+k, 0, other. [B ] j,k. We will fid the qth power (q Nofthematrix(. Theorem relates all positive iteger powers of the matrix B to A ad J. Theorem. If a, b \ (0, ad Nad if B atitridiag (b, a, b,the where A tridiag (b, a, b. (5 B q Aq, if q is eve, J A q, if q is odd, (6 Proof. We will proceed by iductio o q. Thecaseqis obvious. Suppose that the result is true for q ad cosider case q+. By the iductio hypothesis we have B B q B J A q, if q+is eve, B A q, if q+is odd. (7 B q+ Sice B J A we obtai that J A J A q, if q+is eve, J A (A q, if q+is odd. (8 B q+ Sice A is symmetric ad J J,wehave B q Aq, if q is eve, J A q, if q is odd. (9 Next,wehavetosolveA q. We begi this work by reviewig a theorem regardig the Hermitia Toeplitz tridiagoal matrix A. Theorem 3. Let a, b \ (0,ad N.TheA has eigevalues Proof. See [7]. λ i a+ b cos iπ, j. (0 + With the tridiagoal matrix A tridiag (b, a, b, we associate the polyomial sequece P i } i characterized by a three-term recurrece relatio: xp i (x bp i+ (x +ap i (x +bp i (x, i0,,...,. ( With iitial coditios P (x 0 ad P 0 (x,weca write the relatios (imatrixform: xq (x A Q (x +P (x E, ( where Q (x [P 0 (x, P (x,...,p (x] T ad E [0,0,...,0,] T R. Lemma 4. For i 0, the degree of the polyomial P i is i ad P i ad P i+ has o commo root. Proof. See [7]. Oe ca show that the characteristic polyomial of A is precisely (Π c kp (x. Hece the eigevalues of A are exactly the roots of P. If (λ j 0 j are the roots of the polyomial P,the it follows from ( that each λ j is a eigevalue of the matrix A ad Q (x [P 0 (x, P (x,...,p (x] T is a correspodig eigevector [5, 7, 8]. This observatio should betakeitoaccoutelsewhereithepaper. The polyomials P i } 0 i verify the well-kow hristoffel-darboux Idetity.
ISRN omputatioal Mathematics 3 Lemma 5. We have i0 P i (x P i (y P (y P (x P (x P (y y x Proof. See [7]. Tedig y to x i formula (3, we get i0, for x y. (3 P i (x P i (x P (x P (x P (x P (x. (4 Sice the matrix A has distict eigevalues λ 0,λ,...,λ, thus, the eigedecompositio of the matrix A is A TDT, (5 where Ddiag(λ 0,λ,...,λ ad T is the trasformig matrix formed by the eigevectors of A.Namely,T(t P i (λ j,wherep i } i are defied as above. Lemma 6. If T (s,the P j (λ i s P (λ i P (λ i. (6 Proof. By usig the relatios (3ad(4, we obtai s i,k t k,j where δ if ijad δ 0if i j. For q N,wehaveA q TDq T. We get immediately the followig. P k (λ i P k (λ j P (λ δ i P (λ i, (7 Theorem 7. Assume that q Nad A q (α(q.the λ q k P i (λ k P j (λ k P (λ k P (λ k. (8 By usig the auchy Itegral Formula, we ca give aother expressios of the coefficiets as follows: z q P i (z P j (z dz, (9 P (z P (z where is a closed curve cotaiig the roots of P ad o roots of P. orollary 8. If the matrix A is osigular with A,the (α ( α ( A TD T, P i (λ k P j (λ k (0 λ k P (λ k P (λ k. ByusigtheauchyItegralFormula,wecagiveotherexpressios of the coefficiets α ( : α ( z P i (z P j (z dz, ( P (z P (z where is a closed curve cotaiig the roots of P adoroots of P. Theorem 9. Assume that q Nad B q (β(q.the λ q k λ q k P i (λ k P j (λ k P (λ k P (λ k, if q is eve, P i (λ k P j (λ k P (λ k P (λ k, if q is odd. ( ByusigtheauchyItegralFormula,wecagiveotherexpressios of the coefficiets : z q P i (z P j (z P (z P (z dz, z q P i (z P j (z P (z P (z dz, if q is eve, if q is odd, (3 where is a closed curve cotaiig the roots of P adoroots of P. Proof. From Theorem we get [B q ] [A q ], [J A q ] [J ] i,h [A q ] h,j, Namely, [Aq ], if q is eve, [A q ] +, if q is odd. if q is eve, if q is odd. (4, if q is eve, (5 +, if q is odd. From Theorem 7 it follows that λ q k λ q k P i (λ k P j (λ k P (λ k P (λ k, if q is eve, P i (λ k P j (λ k P (λ k P (λ k, if q is odd. (6 By usig the auchy Itegral Formula, we ca give other expressios of the coefficiets : z q P i (z P j (z P (z P (z dz, z q P i (z P j (z P (z P (z dz, if q is eve, if q is odd. (7
4 ISRN omputatioal Mathematics orollary 0. Assume that q Nad B The β ( (β (. P i (λ k P j (λ k λ k P (λ k P (λ k. (8 ByusigtheauchyItegralFormula,wecagiveotherexpressios of the coefficiets β ( : β ( z ( P i (z P j (z dz, (9 P (z P (z where is a closed curve cotaiig the roots of P ad o roots of P. 3. Numerical Examples osider the order atitridiagoal matrix B of the followig type: Assume that B ( ( a a a a c c c a. (30 a a A ( a, d d d a ( a J ( c, (3 where A ad J are matrix. The polyomial sequece P i } i verifies xp i (x P i+ (x +ap i (x +P i (x, i0,,...,, with iitial coditios P (x 0 ad P 0 (x. By simple calculatio we ca show that (3 P i (x U i ( x a, i0,...,, (33 where U i are the hebyshev polyomials [8] ofthesecod kid which satisfies the three-term recurrece relatios: xu i (x U i+ (x +U i (x, (34 with iitial coditios U 0 (x ad U (x x. Each U satisfies si ((+ arccos x U (x, (35 si (arccos x adthustherootsofu (x are z k cos(/( +, k,...,.the,theeigevaluesofaare λ k a+cos,,...,. (36 + We get by Theorem 7 the followig. Assume that q Nad A q (α(q.the (a + cos q + U i (cos (/ (+ U j (cos (/ (+ U (cos (/ (+ U (cos (/ (+. (37 If the matrix A is osigular ad A (α (, the a ( a+cos (/ (+ U i (cos (/ (+ U j (cos (/ (+ U (cos (/ (+ U (cos (/ (+. We ca obtai the followig: β ( (a + cos q + U i (cos (/ (+ U j (cos (/ (+ U (cos (/ (+ U (cos (/ (+, (a + cos q + if q is eve, U i (cos (/ (+ U j (cos (/ (+ U (cos (/ (+ U (cos (/ (+, (a + cos + if q is odd, (38 U i (cos (/ (+ U j (cos (/ (+ U (cos (/ (+ U (cos (/ (+. (39
ISRN omputatioal Mathematics 5 Theorem. osider a odd atural umber m+, m N.LetB atitridiag (,0,ad λ k cos(/( + for every k.the [B q ] +( q+i+j + ( / for all q Nad i, j,where β (k λ q k (4 λ k U i ( λ k U j ( λ k, (40 β (k, if i+jis eve, ( k (4, if i+jis odd. λ k (k,,..., are the eigevalues of the matrix B ad U k (x is the kth degree hebyshev polyomial of the secod kid. Theorem. osider a eve atural umber m, m N.LetB atitridiag (,0,ad λ k cos(/( + for every k.the / [B q ] + γ λ q k (4 λ k U i ( λ k U j ( λ k for all q Nad i, j,where (4 γ, if i+jis eve, (43 0, if i+jis odd. ForeveordermatrixBthefollowigcoditioisfulfilled: λ k 0 (k,4,...,. This meas that eve order matrix B atitridiag (, 0, is osigular (its determiat is ot equal to zero ad derived expressio of B q ca be applied for computig egative iteger powers, as well. Takig q,we get the followig expressio for elemets of the iverse matrix B : [B ] + γ / λ k (4 λ k U i ( λ k U j ( λ k, 0,,...,. (44 Theorem 3. osider a eve atural umber m, m N.LetB atitridiag (,, ad λ k ( k ( + cos(/( +, (k,,...,.the λ q k [B q ] + kiπ kjπ si si + + 4. oclusio ad Discussio I this paper, we derive a geeral expressio for the pth power (p N of ay complex persymmetric atitridiagoal Hakel (costat atidiagoals matrices with costat atidiagoals (atitridiag (b, a, b. This ovel expressio is both a extesio of the oe obtaied by Rimas for the powers of the matrix atitridiag (, 0, with N(see [] fortheoddcasead [5] for the eve case ad a extesio of the oe obtaied by Hogli Wu for the powers of the matrix atitridiag (,, with N(see [3] for the eve case. We may safely draw the coclusio that our results geeralize the results i the related literature [, 3, 5]. oflict of Iterests The author declares that there is o coflict of iterests regardig the publicatio of this paper. Ackowledgmet The author is idebted to the referee for various helpful commets i this paper. Refereces [] J. Gutiérrez-Gutiérrez, Powers of complex persymmetric or skew-persymmetric ati-tridiagoal matrices with costat ati-diagoals, Applied Mathematics ad omputatio, vol. 7, o.3,pp.65 63,0. [] J. Rimas, O computig of arbitrary positive iteger powers for oe type of symmetric ati-tridiagoal matrices of odd order, Applied Mathematics ad omputatio,vol.03,o.,pp.573 58, 008. [3] H. Wu, O computig of arbitrary positive powers for oe type of ati-tridiagoal matrices of eve order, Applied Mathematics ad omputatio,vol.7,o.6,pp.750 756,00. [4] Q. Yi, O computig of arbitrary positive powers for atitridiagoal matrices of eve order, Applied Mathematics ad omputatio,vol.03,o.,pp.5 57,008. [5] J. Rimas, O computig of arbitrary positive iteger powers of odd order ati-tridiagoal matrices with zeros i mai skew diagoal ad elemets,,,, ;,,,, i eighbourig diagoals, Applied Mathematics ad omputatio,vol.0,o.,pp.64 7,009. [6] J. Rimas, Iteger powers of real odd order skew-persymmetric ati-tridiagoal matrices with costat ati-diagoals (atitridiag (a,c, a,a R\0},c R, Applied Mathematics ad omputatio,vol.9,o.,pp.7075 7088,03. [7] R. A. Hor ad. R. Johso, Matrix Aalysis, ambridge Uiversity Press, New York, NY, USA, 990. [8]L.FoxadI.B.Parker,hebyshev Polyomials i Numerical Aalysis, Oxford Uiversity Press, Lodo, UK, 968. [λ q k + + ( i+j (λ k +( k q ] si kiπ kjπ si + +. (45
Advaces i Operatios Research Hidawi Publishig orporatio http://www.hidawi.com Advaces i Decisio Scieces Hidawi Publishig orporatio http://www.hidawi.com Mathematical Problems i Egieerig Hidawi Publishig orporatio http://www.hidawi.com Joural of Algebra Hidawi Publishig orporatio http://www.hidawi.com Probability ad Statistics The Scietific World Joural Hidawi Publishig orporatio http://www.hidawi.com Hidawi Publishig orporatio http://www.hidawi.com Iteratioal Joural of Differetial Equatios Hidawi Publishig orporatio http://www.hidawi.com Submit your mauscripts at http://www.hidawi.com Iteratioal Joural of Advaces i ombiatorics Hidawi Publishig orporatio http://www.hidawi.com Mathematical Physics Hidawi Publishig orporatio http://www.hidawi.com Joural of omplex Aalysis Hidawi Publishig orporatio http://www.hidawi.com Iteratioal Joural of Mathematics ad Mathematical Scieces Joural of Hidawi Publishig orporatio http://www.hidawi.com Stochastic Aalysis Abstract ad Applied Aalysis Hidawi Publishig orporatio http://www.hidawi.com Hidawi Publishig orporatio http://www.hidawi.com Iteratioal Joural of Mathematics Discrete Dyamics i Nature ad Society Joural of Joural of Discrete Mathematics Joural of Hidawi Publishig orporatio http://www.hidawi.com Applied Mathematics Joural of Fuctio Spaces Hidawi Publishig orporatio http://www.hidawi.com Hidawi Publishig orporatio http://www.hidawi.com Hidawi Publishig orporatio http://www.hidawi.com Optimizatio Hidawi Publishig orporatio http://www.hidawi.com Hidawi Publishig orporatio http://www.hidawi.com