INTEGER POWERS OF ANTI-BIDIAGONAL HANKEL MATRICES

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1 Indian J Pure Appl Math, 49: 87-98, March 08 c Indian National Science Academy DOI: 0007/s INTEGER POWERS OF ANTI-BIDIAGONAL HANKEL MATRICES João Lita da Silva Department of Mathematics and GeoBioTec Faculty of Sciences and Technology NOVA University of Lisbon Quinta da Torre, Caparica, Portugal jfls@fctunlpt; joaolita@gmailcom Received 8 September 06; accepted 9 May 07 In this paper we derive a general expression for integer powers of real upper and lower antibidiagonal matrices with constant anti-diagonals using Chebyshev polynomials An explicit formula for the inverse of these matrices is also provided Key words : Hankel matrices; anti-bidiagonal matrices; Chebyshev polynomials INTRODUCTION The computation of integer powers of square matrices is required in several branches of mathematics such as differential equations, linear dynamical systems, graph theory or numerical analysis particularly, it is important in some numerical problems in order to know whether a square matrix is convergent or semi-convergent Recently, some authors studied this subject for a specific type of Hankel matrices see [5-8,, ] among others The main goal of this paper is to establish an expression for integer powers of real anti-bidiagonal Hankel matrices, ie matrices of the form antitridiag n a, c, 0 and antitridiag n 0, c, b, a, b, c R, proceeding with the formulas developed in [6] and [7] Recall that the shape of these matrices, namely the absence of symmetry about the northeast-southwest diagonal in general, makes the approach to this problem be more tricky than the persymmetric case To overcome this nuisance, we shall employ a useful lemma presented in [6] and suitable results about eigenvalues and eigenvectors of general tridiagonal matrices This technique will allow us to achieve the desired expressions for integer powers of both upper and lower real anti-bidiagonal Hankel matrices at the expense of Chebyshev

2 88 JOÃO LITA DA SILVA polynomials Additionally, a formula not depending on any unknown parameter for the inverse of complex anti-bidiagonal Hankel matrices will be also presented INTEGER POWERS: GENERAL EXPRESSION We begin this section by reviewing some special types of matrices that play a central role in this paper We say that an n n matrix is upper anti-bidiagonal if it has the form 0 0 a n c n a n c n 0 c n ; 0 a 3 c 3 a c c 0 0 an n n matrix is said to be lower anti-bidiagonal if it has the form 0 0 c n c n b n c n b n 0 c 3 0 c b 3 c b 0 0 We say that an n n matrix is anti-bidiagonal if it is upper anti-bidiagonal or lower antibidiagonal

3 INTEGER POWERS OF ANTI-BIDIAGONAL HANKEL MATRICES 89 Consider the n n upper anti-bidiagonal matrix with constant anti-diagonals 0 0 a c a c 0 c 0 0 antitridiag n a, c, 0 = 0 a c a c 0 c and the following n n lower bidiagonal matrix c a c 0 0 a c tridiag n a, c, 0 = c 0 0 a c a c Our first auxiliary result relates integer powers of with and the n n exchange matrix J n = The proof can be found in [6] page 34 Lemma Let P n be an n n persymmetric matrix and S n = J n P n where J n is the n n exchange matrix 3 If m is a positive integer then P S m n P n m if m is even, n = J n P n P n P n m if m is odd An n n matrix P n is said persymmetric if it is symmetric about its northeast-southwest diagonal, ie [P n ] k,l = [P n] n l+,n k+ for all k, l see [], page 93

4 90 JOÃO LITA DA SILVA Furthermore, if P n is nonsingular then S m n = P n P n m if m is even, J n P n P n P n m+ if m is odd matrix The next result will permit us to identify the eigenvalues and also eigenvectors of a tridiagonal c b 0 0 a c b 0 a 3 c 3 M n = c n b n 0 a n c n b n 0 0 a n c n where {a k } k n, {b k } k n and {c k } k n are sequences of complex numbers such that b k 0 and b n = Supposing the polynomial sequence {Q k x} k 0 characterized by a three-term recurrence relation 4 xq 0 x = c Q 0 x + b Q x xq k x = a k Q k x + c k Q k x + b k Q k x for k 5 with initial condition Q 0 x =, we can give a matrix form for this three-term recurrence relation: xqx = M n qx + Q n xe 6 where M n is given by 4, qx = Q 0 x Q x Q n x and e = 0 0 The following statement is a well-known result Lemma If λ is a zero of the polynomial Q n x obtained by 5 then λ is an eigenvalue of

5 INTEGER POWERS OF ANTI-BIDIAGONAL HANKEL MATRICES 9 the tridiagonal matrix 4 and qλ = Q 0 λ Q λ Q n λ is a corresponding eigenvector PROOF : Let λ be a zero of the polynomial Q n x From 6 we have M n qλ = λqλ and λ is an eigenvalue of M n Since Q 0 x 0 it follows also that qλ is a eigenvector associated to λ Setting P n = tridiag n a, c, 0 with ac 0 we have S n = J n P n = antitridiag n a, c, 0 and P n P n = a + c ac 0 0 ac a + c ac 0 ac a + c a + c ac 0 ac a + c ac 0 0 ac c = ac

6 9 JOÃO LITA DA SILVA where = a c Hence, integer powers of 0 0 a c a c 0 c 0 0 antitridiag n a, c, 0 = 0 a c a c 0 c can be computed at the expense of the integer powers of the following tridiagonal n n matrix T n = Let us observe that the case c = 0 is already established in [6] page 36, as well as the case a = 0 which was presented in formulae 3a, 3b, 3c and 3d of [7] In [3], the authors established a necessary condition for the eigenvalues of tridiagonal Toeplitz matrices having four corners perturbed giving also the corresponding eigenvectors, and their results can be used on matrix T n However, our approach involving Chebyshev polynomials takes advantage of many results stated in the scope of the theory of orthogonal polynomials, particularly, numerical methods developed by Grant and Rahman to determine the zeros of a linear combination of Chebyshev polynomials see [3] and [4], for details Denoting by U p x, p 0 the pth degree Chebyshev polynomial of the second kind U p x = sin[p + arccos x], < x < sinarccos x with U p ± = ± p p + see [9], the polynomial sequence {Q k x} k 0 defined in 5 for the matrix 8 will be x Q k x = U k, 0 k n 7

7 INTEGER POWERS OF ANTI-BIDIAGONAL HANKEL MATRICES 93 x and Q n x = U n +U x n It is well-known that the zeros of Q n x are all real and simple see, page 7 Thus, setting θ k, k =,, n as the solutions of the equation U n t + U n t = 0, 9 the result below will provide us important matrices in the computing process of integer powers of real anti-bidiagonal Hankel matrices As pointed out above, the solutions θ k can be obtained, at least, using numerical methods see [3] and [4] Lemma 3 Let n N and a, c R \ {0} If [ U n ]k,l = U k θ l for all k, l =,, n with θ j, j =,, n the solutions of 9 then U n U n = diag [U n θ + a c U n θ ] U n θ,, [ U n θ n + a c U n θ n ] U n θ n PROOF : Setting = a c we have [ U n U n ]k,l = n Q j + + θ k Q j + + θ l = n U j θ k U j θ l j=0 where θ k, k =,, n are the solutions of 9 From Christoffel-Darboux identities see [], pages 3 and 4 we obtain n Q j + + θ k Q j + + θ l = Q n+ +θ kq n + +θ l Q n + +θ kq n + +θ l θ k θ l j=0 j=0 = [Unθ k+u n θ k ]U n θ l U n θ k [U nθ l +U n θ l ] θ k θ l = 0 for each k, l =,, n such that k l since from 9 we have U n θ k + U n θ k = 0 for all k =,, n On the other hand, using the confluent form of the Christoffel-Darboux identity, n Q j j=0 + + θ k = [ U n θ k + U n θ k ] U n θ k U n θ k U n θ k = [ U n θ k + U n θ k ] U n θ k for each k =,, n, which establishes the thesis Remark : The values involving U px can be computed using the Chebyshev polynomials of the second kind since U px = p+u p x pu p+ x x if p > 0 0 if p = 0

8 94 JOÃO LITA DA SILVA and U p± = ± p+ p+ 3 p+ 3 whenever p > 0 Moreover, the matrix U n of Lemma 3 is nonsingular since [ U n θ k + a c U n θ k ] U n θ k > 0 for all k =,, n see, page 4, yielding U n = diag,, [U nθ + a c U n θ ]U n θ [U nθ n + a c U n θ n]u n θ n U n The result below will give us positive integer powers of real anti-bidiagonal Hankel matrices Theorem Let a, c R \ {0} and n, m N If U n = [ U k θ l ], k, l =,, n k,l and Θ n = diag a c + c a + θ,, a c + c a + θ n with θj, j =,, n the solutions of 9 then U n Θ n U n is an eigenvalue decomposition of the n n matrix T n in 8 Moreover, if A n := antitridiag n a, c, 0 then ac m Un Θ m n D n U A m n if m is even, n = ac m A n U n Θ m n D n U n if m is odd where D n = diag,, [U n θ + a c U n θ ]U n θ [U n θ n+ a c U n θ n]u n θ n PROOF : Let = a c and θ, θ,, θ n the solutions of 9 From Lemma each + + θ k, k =,, n is an eigenvalue of T n and Q θ k U 0 θ k Q + u k = + θ k U θ k = Q n + + θ k U n θ k is a corresponding non-null eigenvector of T n associated to + + θ k Hence, {u, u,, u n } is a complete set of eigenvectors of T n see [0], page 507 leading to U n = [u u u n ] and T n = U n Θ n U n The last sentence follows directly from Lemma 3 and subsequent Remark Remark : The formulas announced in Theorem can be extended to any negative integer m since deta n = detj n det [tridiag n a, c, 0] = c n nn 0 From Theorem, we can now compute the entries of any integer power of a real upper antibidiagonal Hankel matrix using the solutions of 9 and Chebyshev polynomials of the second kind The main result of this paper follows next

9 INTEGER POWERS OF ANTI-BIDIAGONAL HANKEL MATRICES 95 Theorem Let a, c R \ {0} and n, m N If A n := antitridiag n a, c, 0 and θ j, j =,, n are the solutions of 9 then [ A ±m n ]k,l = ac± m n a c + c a +θ j ± m U [U nθ j + a c U n θ j]u n θ j k θ j U l θ j 0 j= when m is even and [ A ±m n ]k,l = ac ±m cac ±m n j= n j= whenever m is odd for all k, l =,, n a c + c a +θ j ±m [au [U nθ j + a c U n θ j]u n θ j n k θ j + cu n k θ j ] U l θ j if k < n, a c + c a +θ j ±m U [U nθ j + a c U n θ j]u n θ j l θ j if k = n With the aid of Sherman-Morrison formula, we shall be able to give an explicit expression for the inverse of antitridiag n a, c, 0 Theorem 3 Let n N and a, c C\{0} If A n = antitridiag n a, c, 0, a c + c a + cos kπ n+ n 0 for all k =,, n and sin n+ kπ n+ a c + c kπ + cos a n+ c a then k= A n = ac A n V n B n V n where [V n ] k,l := n+ sin klπ n+ and [B n ] k,l := δ k,l a c + c kπ + cos a n+ + a k+l sin n+ kπ sin n+ lπ [ ] n c a sin n+ jπ n+ a j= c + a c + cos n+ jπ n+[ a c + c kπ + cos a n+][ a c + c lπ + cos a n+] with δ k,l the Kronecker delta PROOF : Suppose n N and a, c C \ {0} Setting

10 96 JOÃO LITA DA SILVA [ ] n := diag a c + c a + cos π n+,, a c + c a + cos nπ n+, a c + c a 0 0 T n := a c + c a a 0 c + c a a c + c a 0 a c + c a a 0 0 c + c a and Ê n := a c we obtain V n T n V n = V n Tn + Ên V n = n a c ww where w = n+ sin π n n+ n+ sin π n+ n+ sin nπ n+ and T n is the matrix 8 Since V n satisfies Vn = I n and V n = Vn, it follows [ ] T n = V n n + a c a c w n w n ww n V n according to the Sherman-Morrison formula see [0], page 4 Thus, Lemma implies { [ A n = J n tridiag n a, c, 0V n ac n + a c a c w n w n ww n ]} V n where J n is the exchange matrix 3 and is established The proof is complete

11 INTEGER POWERS OF ANTI-BIDIAGONAL HANKEL MATRICES 97 Remark : Note that the decomposition T n = V n n a c ww V n leads to ac m A m V n n a c ww m V n if m is even, n = ac m antitridiag n a, c, 0V n n a m c ww V n if m is odd and A m n = ac m V n n a c ww m V n if m is even, ac m+ antitridiag n a, c, 0V n n a c ww m+ V n if m is odd for any positive integer m Furthermore, it is possible to express the eigenvalues of n a c ww as the zeros of explicit rational functions and the corresponding eigenvectors, which allows us to perform an approach similar to the one presented in [8] Thereby, an alternative procedure to compute the integer powers of A n = antitridiag n a, c, 0 is also founded The formulae established throughout for integer powers of real upper anti-bidiagonal Hankel matrices can be used to compute also integer powers of real lower anti-bidiagonal Hankel matrices, 0 0 c c b c b 0 antitridiag n 0, c, b = c 0 c b c b 0 0 Indeed, antitridiag n 0, c, b = J n antitridiag n b, c, 0J n, so that, for any integer m we have [antitridiag n 0, c, b] m = J n [antitridiag n b, c, 0] m J n, ie {[antitridiag n 0, c, b] m } k,l = {[antitridiag n b, c, 0] m } n k+,n l+ and similar formulas to 0, and can be obtained for lower anti-bidiagonal Hankel matrices ACKNOWLEDGEMENT This work is a contribution to the Project UID/GEO/04035/03, funded by FCT - Fundação para a Ciência e a Tecnologia, Portugal

12 98 JOÃO LITA DA SILVA REFERENCES T S Chihara, An introduction to orthogonal polynomials, Gordon and Breach, New York 978 G H Golub and C F Van Loan, Matrix computations, The Johns Hopkins University Press, Baltimore, MD J A Grant and A A Rahman, Determination of the zeros of a linear combination of Chebyshev polynomials, IMA J Numer Anal, 3 983, J A Grant and A A Rahman, Determination of the zeros of a linear combination of generalised polynomials, J Comput Appl Math, 4 99, J Gutiérrez-Gutiérrez, Powers of complex persymmetric or skew-persymmetric anti-tridiagonal matrices with constant anti-diagonals, Appl Math Comput, 7 0, J Lita da Silva, Integer powers of anti-tridiagonal matrices of the form antitridiag n a, 0, b, a, b R, Comput Math Appl, 69 05, J Lita da Silva, Integer powers of anti-tridiagonal matrices of the form antitridiag n a, c, a, a, c C, Int J Comput Math, , J Lita da Silva, On anti-pentadiagonal persymmetric Hankel matrices with perturbed corners, Comput Math Appl, 7 06, J C Mason and D Handscomb, Chebyshev polynomials, Chapman & Hall/CRC, Boca Raton FL, C D Meyer, Matrix analysis applied linear algebra, Society for Industrial and Applied Mathematics, Philadelphia, PA 000 J Rimas, Integer powers of real odd order skew-persymmetric anti-tridiagonal matrices with constant anti-diagonals antitridiag n a, c, a, a R \ {0}, c R, Appl Math Comput, 9 03, J Rimas, Integer powers of real even order anti-tridiagonal Hankel matrices of the form antitridiag n a, c, a, Appl Math Comput, 5 03, W C Yueh and S S Cheng, Explicit eigenvalues and inverses of tridiagonal Toeplitz matrices with four perturbed corners, ANZIAM J, ,