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Spectral Properties of Anti-Heptadiagonal Persymmetric

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Spectral Properties of Anti-Heptadiagonal Persymmetric Spectral properties of anti-heptadiagonal persymmetric Hankel matrices Jo˜ao Lita da Silva1 Department of Mathematics and GeoBioTec Faculty of Sciences and Technology NOVA University of Lisbon Quinta da Torre, 2829-516 Caparica, Portugal Abstract In this paper we express the eigenvalues of anti-heptadiagonal persymmetric Hankel matrices as the zeros of explicit polynomials giving also a representation of its eigenvectors. We present also an expression depending on localizable parameters to compute its integer powers. In particular, an explicit formula not depending on any unknown parameter for the inverse of anti-heptadiagonal persymmetric Hankel matrices is provided. Key words: Anti-heptadiagonal matrix, Hankel matrix, eigenvalue, eigenvector, diagonalization 2010 Mathematics Subject Classification: 15A18, 15B05 1 Introduction The importance of Hankel matrices in computational mathematics and engineering is well-known. As a matter of fact, these type of matrices have not only a varied and numerous relations with polyno- mial computations (see, for instance, [2]) but also applications in engineering problems of system and control theory (see [4], [12] or [14] and the references therein). Recently, several authors have studied particular cases of these matrices in order to derive explicit expressions for its powers (see [7], [11], [16], [17], [19], [20] among others). The aim of this paper is to explore spectral properties of general anti-heptadiagonal persymmetric Hankel matrices, namely locating its eigenvalues and getting an explicit representation of its eigenvec- tors. Additionally, it is our purpose announce formulae for the computation of its integer powers using, thereunto, a diagonalization for the referred matrices. Particularly, an expression free of any unknown parameter to calculate the inverse of any anti-heptadiagonal persymmetric Hankel matrix (assuming arXiv:1907.00260v1 [math.RA] 29 Jun 2019 its nonsingularity) is made available. The essential ingredient in the present approach is a suitable decomposition for these sort of matrices, obtained at the expense of the so-called bordering technique originally settled in the eighties for band symmetric Toeplitz matrices. Thereby, the statements an- nounced here besides being a small contribution for the state of art on spectral features of anti-banded matrices, show that techniques developed for band symmetric Toeplitz matrices can be also employed in persymmetric Hankel matrices. We emphasize that, in general, the acquaintance of the eigenvalues and eigenvectors of a symmetric Toeplitz matrix does not allow to know which are exactly the eigen- values and eigenvectors of the corresponding persymmetric Hankel matrix (obtained by multiplication of the exchange matrix), despite multiple symmetries lead to some relationship between eigenpars of these two classes of matrices. 1E-mail address: [email protected]; [email protected] 1 An n × n matrix Bn is said to be banded when all its nonzero elements are confined within a band formed by diagonals parallel to the main diagonal, i.e. [Bn]k,ℓ = 0 when |k − ℓ| >h, and [Bn]k,k−h 6=0 or [Bn]k,k+h 6= 0 for at least one value of k, where h is the half-bandwidth and 2h + 1 is the bandwidth (see [15], page 13). An n × n matrix An is said to be anti-banded if JnAn is a banded matrix where Jn is the n × n exchange matrix (that is, the matrix such that [Jn]k,n−k+1 = 1 for each 1 6 k 6 n with the remaining entries zero). Particularly, an n × n matrix which has the form 0 ............... 0 αn βn γn dn . .. . αn−1 βn−1 γn−1 dn−1 cn−1 . . .. .. β γ d c b n−2 n−2 n−2 n−2 n−2 . . .. .. .. . γn−3 dn−3 cn−3 bn−3 an−3 . . .. .. .. .. . dn−4 cn−4 bn−4 an−4 0 . . .. .. .. .. .. .. .. .. .. . .. .. .. .. 0 α5 β5 γ5 d5 . .. .. .. α4 β4 γ4 d4 c4 . .. .. β3 γ3 d3 c3 b3 . .. γ2 d2 c2 b2 a2 . d1 c1 b1 a1 0 ............... 0 is called anti-heptadiagonal. Throughout, we shall consider the following n × n anti-heptadiagonal persymmetric Hankel matrix 0 ............... 0 a b cd . . .. a b c dc . . .. .. b c d cb . . .. .. .. . c d c ba . . .. .. .. .. . d c b a 0 . Hn = . . (1.1) . .. .. .. .. 0 a b c d . .. .. .. ab c d c . .. .. bc d c b . . cd c b a .. dc b a 0 ............... 0 2 Auxiliary tools Consider the class of complex matrices defined by A M C n := An ∈ n( ):[An]k−1,ℓ + [An]k+1,ℓ = [An]k,ℓ−1 + [An]k,ℓ+1 , k,ℓ =1, 2,...,n n o where it is assumed [An]n+1,ℓ = [An]k,n+1 = [An]0,ℓ = [An]k,0 = 0 for all k,ℓ. We begin by retrieve a pair of results stated in [1] (see Propositions 2.1 and 2.2), announced here for complex matrices. The proofs do not suffer any changes from the original ones, whence we will omit the details. 2 Lemma 1 (a) The class An is the algebra generated over C by the n × n matrix 0 1 0 ......... 0 .. 1 0 1 . .. .. 0 1 0 . . Ωn = . .. .. .. .. .. . (2.1) . . .. .. 0 1 0 . . .. 1 0 1 0 ......... 0 1 0 ⊤ (b) If An ∈ An and a is its first row then n−1 k An = ωk+1Ωn k X=0 ⊤ where Ωn is the n × n matrix (2.1) and ω = (ω1,ω2,...,ωn) is the solution of the upper triangular ω system Tn = a, with [Tn]0,0 := 1, [Tn]0,ℓ := 0 for all 1 6 ℓ 6 n, [T ] + [T ] , 1 6 k 6 ℓ 6 n [T ] := n k−1,ℓ−1 n k+1,ℓ−1 . n k,ℓ 0, otherwise Given a positive integer n, Sn+2 will denote the (n + 2) × (n + 2) symmetric, involutory and orthogonal matrix defined by 2 kℓπ [S ] := sin . (2.2) n+2 k,ℓ n +3 n +3 r Our second auxiliary result makes useful Lemma 1 above and provides us an orthogonal diagonalization for the following (n + 2) × (n + 2) complex matrix 0 ............... 0 a b c − a d − b . . .. abc dc − a . . .. .. bcd c b . . .. .. .. . cdc b a . . .. .. .. .. . dcb a 0 . Hn = . . (2.3) +2 . .. .. .. .. b 0 a bcd . .. .. .. a b cdc . .. .. b c dcb . . c − ad cba .. d − b c − a b a 0 ............... 0 Lemma 2 Let n ∈ N, a,b,c,d ∈ C and nkπ (n+1)kπ (n+2)kπ λk = −2a cos n+3 − 2b cos n+3 − 2c cos n+3 − d cos(kπ) , k =1, 2,...,n +2. (2.4) h i h i 3 If Hn+2 is the (n + 2) × (n + 2) matrix (2.3) then b Hn+2 = Sn+2Λn+2Sn+2 where Λn+2 = diag (λ1, λ2,...,λn+2) andb Sn+2 is the (n + 2) × (n + 2) matrix (2.2). Proof. Suppose n ∈ N and a,b,c,d ∈ C. Since Hn+2 ∈ An+2 and its first row is ⊤ a = (0,...,b0,a,b,c − a, d − b) we have, from Lemma 1, 2 3 Hn+2 = (d − 2b)In+2 + (c − 3a)Ωn+2 + b Ωn+2 + a Ωn+2 Jn+2 where Jn+2 is the (bn + 2) × (n + 2) exchange matrix (that is, the matrix such that [Jn+2]k,n−k+3 =1 for every 1 6 k 6 n + 2 with the remaining entries zero). Using the spectral decomposition n+2 ℓπ Ω = 2cos s s⊤, n+2 n +3 ℓ ℓ ℓ X=1 where 2 ℓπ n+3 sin n+3 q 2 2ℓπ n+3 sin n+3 sℓ = q . . 2 (n+1)ℓπ n sin n +3 +3 q h i (i.e. the ℓth column of Sn+2 in (2.2)) and the decomposition 2 3 n+3 Jn+2 = Sn+2 diag (−1) , (−1) ,..., (−1) Sn+2 (see Lemma 1 of [8]) with Sn+2 given by (2.2), it follows n+2 ℓπ 2 ℓπ 3 ℓπ ⊤ Hn+2 = (d − 2b)+2(c − 3a)cos +4b cos +8a cos sℓ sℓ Jn+2 ( n +3 n +3 n +3 ) ℓ=1 . X 2 3 n+3 2 3 n+3 b = Sn+2 diag (−1) λ1, (−1) λ2,..., (−1) λn+2 diag (−1) , (−1) ,..., (−1) Sn+2 = Sn+2 diag (λ1, λ2,...,λn+2) Sn+2 The proof is completed. The decomposition below plays a central role in the study of the spectral properties of anti- heptadiagonal persymmetric Hankel matrices. The statement is split in two cases of n ∈ N: n is an even number and n is an odd number. For the case in which n is even, we shall take the same steps as in [1] (see pages 106 and 107); the case in which n is odd, this procedure must be improved and we shall follow closely the bordering technique of Fasino (see [5], pages 303 and 304). Lemma 3 Let n ∈ N, a,b,c,d ∈ C, λk, k =1, 2,...,n +2 be given by (2.4) and Hn the n × n matrix (1.1). (a) If n is even then ⊤ ⊤ diag (λ3, λ5,...,λn+1)+ λ1uu O Hn = RnPn ⊤ PnRn (2.5a) O diag (λ , λ ,...,λn)+ λn vv 2 4 +2 4 where 3π 2π sin( n+3 ) sin( n+3 ) π π sin( n+3 ) sin( n+3 ) 5π 4π sin( n+3 ) sin( n+3 ) π π sin( n ) sin( n ) u = +3 , v = +3 , (2.5b) . . . . (n+1)π nπ sin[ n+3 ] sin( n+3 ) π π sin( n ) sin( n ) +3 +3 Rn is the n × n matrix given by 2 (k + 1)(ℓ + 1)π [R ] = sin (2.5c) n k,ℓ n +3 n +3 r and Pn is the n × n permutation matrix defined by 1 if ℓ =2k − n − 1 or ℓ =2k [Pn]k,ℓ = .
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