Orthogonal diagonalization for complex skew-persymmetric anti-tridiagonal Hankel matrices

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1 Spec. Matrices 016; 4:73 79 Research Article Open Access Jesús Gutiérrez-Gutiérrez* and Marta Zárraga-Rodríguez Orthogonal diagonalization for complex skew-persymmetric anti-tridiagonal Hankel matrices DOI /spma Received May 13, 015; accepted November 1, 015 Abstract: In this paper, we obtain an eigenvalue decomposition for any complex skew-persymmetric antitridiagonal Hankel matrix where the eigenvector matrix is orthogonal. Keywords: Anti-tridiagonal matrices; Hankel matrices; orthogonal diagonalization; skew-persymmetric matrices. 1 Introduction An n n complex anti-tridiagonal Hankel matrix is a matrix of the form antitridiag n (a 1, a 0, a 1 ) : 0 0 a 1 a a 1 a 0 a a 1 a 0 a a a 0 a 1 0 0, n N, a 1, a 0, a 1 C, where N is the set of natural numbers (that is, the set of positive integers) and C denotes the set of (finite) complex numbers. In [1, Theorem 4] an orthogonal diagonalization for any real persymmetric anti-tridiagonal Hankel matrix was presented. The following result, which was given in [, Theorem 5 and Lemma 1], is a generalization of [1, Theorem 4] to complex matrices ([3] is another recent reference where the eigenvalues of certain persymmetric anti-tridiagonal Hankel matrices are given). Theorem 1. Let n N and a 1, a 0 C. Then antitridiag n (a 1, a 0, a 1 ) Υ n diag(τ 1,..., τ n )Υ 1 n, where diag(τ 1,..., τ n ) : (τ j δ ) n 1, with δ being the Kronecker delta and ) τ j ( 1) ( a j+1 jπ 0 + a 1, j {1,..., n}, *Corresponding Author: Jesús Gutiérrez-Gutiérrez: Ceit, Manuel Lardizábal 15, 0018 San Sebastián, Spain and Tecnun (University of Navarra), Manuel Lardizábal 13, 0018 San Sebastián, Spain, jgutierrez@ceit.es Marta Zárraga-Rodríguez: ISSA (University of Navarra), Ediffcio Amigos, Campus Universitario, Pamplona, Spain and Tecnun (University of Navarra), Manuel Lardizábal 13, 0018 San Sebastián, Spain, mzarraga@tecnun.es 016 Jesús Gutiérrez-Gutiérrez and Marta Zárraga-Rodríguez, published by De Gruyter Open. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs 3.0 License. Download Date 4/10/18 :13 PM

2 74 Jesús Gutiérrez-Gutiérrez and Marta Zárraga-Rodríguez and Υ n is the n n real symmetric orthogonal matrix whose entries are given by [Υ n ] jkπ sin, j, k {1,..., n}. In the present paper we give an orthogonal diagonalization for any complex skew-persymmetric antitridiagonal Hankel matrix, i.e., for any matrix of the form antitridiag n (a, 0, a) with a C and n N. Orthogonal diagonalization We first give an orthogonal diagonalization for any complex skew-persymmetric anti-tridiagonal Hankel matrix of odd order. Theorem. Consider a C and n p + 1 with p N {0}. Let Υ n be as in Theorem 1. Then antitridiag n (a, 0, a) U n diag(λ 1,..., λ n )U 1 n, where λ j a and U n is the n n real orthogonal matrix given by jπ, j {1,..., n}, [U n ] s [Υ n ], j, k {1,..., n}, with Proof. See A. s { ( 1) j+1 if j is odd, ( 1) j+1+n+k if j is even, j, k {1,..., n}. (1) We now give an orthogonal diagonalization for any complex skew-persymmetric anti-tridiagonal Hankel matrix of even order. Theorem 3. Consider a C and n p with p N. Let Υ n be as in Theorem 1. Then antitridiag n (a, 0, a) U n diag(λ 1,..., λ n )U 1 n, where λ j a and U n is the n n real orthogonal matrix given by jπ, j {1,..., n}, [U n ] s [Υ n ], j, k {1,..., n}, with s ( 1) j 1 +( 1) n+k+j+1 if j is odd, ( 1) j +( 1) n+k+j if j is even, j, k {1,..., n}. () Proof. See B. In [4, 5] Rimas gave a diagonalization for the matrix antitridiag n (1, 0, 1) of the form ( antitridiag n (1, 0, 1) W n diag π, π,..., nπ ) W 1 n, (3) Download Date 4/10/18 :13 PM

3 Orthogonal diagonalization 75 where the eigenvector matrix W n is real and not orthogonal. Although W n is not orthogonal it satisfies [W n W n ] 0 whenever j, k {1,..., n} and j k, because eigenvectors of a real symmetric matrix corresponding to different eigenvalues are orthogonal (see, e.g., [6, p. 548]). From (3) we have antitridiag n (a, 0, a) W n diag (λ 1,..., λ n) W 1 n (4) for all a C. Thus, the difference between the diagonalization in (4) and the one given in Theorems and 3 for the matrix antitridiag n (a, 0, a) lies in the eigenvector matrix. Since the algebraic multiplicity of λ k is 1 with a 0, its geometric multiplicity is also 1 (see, e.g., [7, Section 4.1]), and consequently, there exists r k R such that [U n ] 1:n,k r k [W n ] 1:n,k, k {1,..., n}, where [U n ] 1:n,k and [W n ] 1:n,k denote the kth column of the matrix U n and W n, respectively. Therefore, 1 [I n ] k,k [U n U n ] k,k [U n ] 1:n,k[U n ] 1:n,k r k[w n ] 1:n,k[W n ] 1:n,k r k [W n] 1:n,k, k {1,..., n}, and hence, r k 1 [W n ] 1:n,k, k {1,..., n}, where denotes the Euclidean norm. Thus, { [Wn ] [U n ] 1:n,k 1:n,k, [W } n] 1:n,k, [W n ] 1:n,k [W n ] 1:n,k k {1,..., n}. (5) Observe that to prove (5) is another way to prove Theorems and 3. We finish by mentioning that Theorems and 3 are useful to obtain the natural powers of the considered matrix antitridiag n (a, 0, a): ( antitridiagn (a, 0, a) ) q Un diag(λ 1 q,..., λ n q )U n, q N. The natural powers of complex skew-persymmetric anti-tridiagonal Hankel matrices were also studied in [4, 5, 8, 9]. In [8] the expression for those powers was obtained by using the eigenvalue decomposition of the complex tridiagonal Toeplitz matrices presented in [10]. Acknowledgement: The authors thank the two anonymous reviewers for their helpful comments. This work was supported in part by the Spanish Ministry of Economy and Competitiveness through the RACHEL project (TEC C4--R). A Proof of Theorem We divide the proof into two steps. In the first step we prove that U n is an orthogonal matrix, or equivalently, that Un U n I n, where denotes transpose and I n is the n n identity matrix. For convenience we rewrite (1) as ) s ( 1) j ( 1) 1+n+k j+1, j, k {1,..., n}, where x denotes the smallest integer not less than x. Applying the basic trigonometric formula sin x sin y (x + y) (x y) (see, e.g., [11, p. 97]) yields [ U n U n ] Un [U n] h,k j,h [U n] h,j [U n] h,k Download Date 4/10/18 :13 PM

4 76 Jesús Gutiérrez-Gutiérrez and Marta Zárraga-Rodríguez ( 1) h ( 1) 1+n+j ) h+1 [Υn ] h,j ( 1) h ) ( 1) 1+n+k h+1 [Υn ] h,k ( 1) j+k) h+1 hjπ hkπ (( 1) j+k ) h+1 1 ( 1)j+k+1 (( 1) j+k ) h ( (j k)hπ ) (j k)hπ, j, k {1,..., n}. Consequently, using and ( 1) h mhπ mhπ 0 if m is odd, n if m Ω, 1 if m N \ Ω, 0 if + m is odd, n if + m Ω and m Ω, n 1 if + m Ω and m N \ Ω, n 1 if + m N \ Ω and m Ω, 1 if + m N \ Ω and m N \ Ω, where m N {0} and Ω {()k : k N {0}} (see [1, Eqs. (5) and (7)]), we obtain 1 ] [ Un [ 1 n] 1 if j k, U n 1 [ 1 ( 1)] 0 if j k and j + k is even, 1 [0 0] 0 if j k and j + k is odd, [I n ], j, k {1,..., n}. (6) (7) In the second step we prove that antitridiag n (a, 0, a) U n D n U 1 n, or equivalently, antitridiag n (a, 0, a) U n D n U n, where D n diag(λ 1,..., λ n ). Applying the basic trigonometric formulas sin x sin y (x + y) (x y) and (x + y) x y sin x sin y (see, e.g., [11, pp ]) yields hπ 1 ( (j+k)hπ (j k)hπ ) 1 (j+k)hπ hπ (j k)hπ hπ 1 ( 1 (j+k+1)hπ + (j+k 1)hπ ) 1 ( (j k+1)hπ + (j k 1)hπ ) 1 (j+k+1)hπ + (j+k 1)hπ (j k+1)hπ (j k 1)hπ 4 (8) for all h, j, k {1,..., n}, and hence, [ U n D n U n 4at at ] [U n] j,h [ D n Un ] h,k [U n] j,h [D n] h,l [ Un ] l,k [U n] j,h [D n] h,l [U n] k,l [U n] j,h [D n] h,h [U n] k,h 4a ( 1) j + k ( 1) j+k) h l1 ( 1) 1+n+h ) j+k l1 ( 1) j+k) h (j + k + 1)hπ (j + k 1)hπ (j k + 1)hπ + (j k 1)hπ, Download Date 4/10/18 :13 PM

5 Orthogonal diagonalization 77 ) where t ( 1) j j+k + k ( 1) 1+n and j, k {1,..., n}. Therefore, using (6) and (7) we conclude that U n D n Un [ ] n + ( 1) ( 1) ( 1) at if j + k n, [ ] ( 1) + n ( 1) ( 1) at if j + k n +, [ ] 0 if j + k is even, [ ] ( 1) + ( 1) ( 1) ( 1) 0 otherwise, at at at at a( 1) j + k+1 1+n ( 1) a( 1) a if j + k n and j is even, a( 1) j+1 + k 1+n ( 1) a( 1) a if j + k n and j is odd, a( 1) j + k+1 1+n ( 1) a( 1) n+ a if j + k n + and j is even, a( 1) j+1 + k ( 1) 1+n a( 1) n+ a if j + k n + and j is odd, 0 otherwise, [antitridiag n (a, 0, a)], j, k {1,..., n}. B Proof of Theorem 3 We divide the proof into two steps. In the first step we prove that U n is an orthogonal matrix. For convenience we rewrite () as ) ( 1) j ( 1) j 1 + ( 1) n+j+k s, j, k {1,..., n}. Applying (6) and (7) yields [ Un U n ] [U n] h,j [U n] h,k 1 ( 1) h ( 1) h 1 + ( 1) n+h+j ( 1) n+h+j 1 + ( 1)j+k () 1 + ( 1) n+h+j ) ) ) 1 + ( 1) n+h+k [Υ n] h,j [Υ n] h,k ) ( 1) h ( 1) h 1 + ( 1) n+h+k [Υ n] h,j [Υ n] h,k ) ) 1 + ( 1) n+h+k sin hjπ hkπ sin [ 1 + ( 1) j+k) + ( 1) h ( 1) n ( 1) j + ( 1) k) ] 1 ( ( 1) h (j k)hπ ( 1) n ( 1)j + ( 1) k (j k)hπ () 1 [ 1 n] ( 1) n ( 1)j [0 0] 1 if j k, 1 [ 1 ( 1)] ( 1) n ( 1)j +( 1) k [0 0] 0 if j k and j + k is even, () if j k and j + k is odd, [I n ], j, k {1,..., n}. (j k)hπ ) Download Date 4/10/18 :13 PM

6 78 Jesús Gutiérrez-Gutiérrez and Marta Zárraga-Rodríguez In the second step we prove that antitridiag n (a, 0, a) U n D n Un 1, or equivalently, antitridiag n (a, 0, a) U n D n Un, where D n diag(λ 1,..., λ n ). We have U n D n Un [U n] j,h [D n] h,h [U n] k,h ( 1) j + k ( 1) j+k ( 1) j + k ( 1) j+k ( 1) j + k ( 1) j+k at 1 + ( 1) j+k + ( 1) h ( 1) n 1 + ( 1) n+j+h ) 1 + ( 1) n+k+h ( 1) j + ( 1) k) ) λ h [Υ n ] j,h [Υ n ] k,h ) λ h [Υ n ] j,h [Υ n ] k,h ( ) 1+( 1) j+k) λ h [Υ n ] j,h [Υ n ] k,h +( 1) n ( 1) j +( 1) k) ( 1) h λ h [Υ n ] j,h [Υ n ] k,h ar where t ( 1) j + k (1 + ( 1) j+k ), r ( 1) j + k ( 1) n (6), (7), and (8) we conclude that U n D n Un at (j+k+1)hπ ( 1) h + (j+k 1)hπ, ( 1) j + ( 1) k), and j, k {1,..., n}. Using (j k+1)hπ (j k 1)hπ () + ar ( 1) h (j+k+1)hπ + (j+k 1)hπ (j k+1)hπ (j k 1)hπ () at () [ ] + ar [ ] ar () n + ( 1) ( 1) ( 1) if j + k n, at () [ ] + ar [ ] ar () 1 + n ( 1) ( 1) if j + k n +, 0 if j + k is odd, at () [ ] + ar [ ] () 1 + ( 1) ( 1) ( 1) 0 otherwise, a ( 1) j + k n ( 1) a( 1) n a if j + k n and j is even, j+1 a ( 1) + k+1 n ( 1) ( ) a( 1) a if j + k n and j is odd, a ( 1) j + k n ( 1) a( 1) a if j + k n + and j is even, j+1 a ( 1) + k+1 n ( 1) ( ) a( 1) n+ a if j + k n + and j is odd, 0 otherwise, [antitridiag n (a, 0, a)], j, k {1,..., n}. References [1] J. Gutiérrez-Gutiérrez, Powers of real persymmetric anti-tridiagonal matrices with constant anti-diagonals, Applied Mathematics and Computation 06 (008) [] J. Gutiérrez-Gutiérrez, Eigenvalue decomposition for persymmetric Hankel matrices with at most three non-zero antidiagonals, Applied Mathematics and Computation 34 (014) [3] M. Akbulak, C. M. da Fonseca, F. Yılmaz, The eigenvalues of a family of persymmetric anti-tridiagonal -Hankel matrices, Applied Mathematics and Computation 5 (013) [4] J. Rimas, On computing of arbitrary integer powers of even order anti-tridiagonal matrices with zeros in main skew diagonal and elements 1, 1, 1,..., 1; 1, 1, 1,..., 1 in neighbouring diagonals, Applied Mathematics and Computation 04 (008) [5] J. Rimas, On computing of arbitrary positive integer powers of odd order anti-tridiagonal matrices with zeros in main skew diagonal and elements 1, 1, 1,..., 1; 1, 1, 1,..., 1 in neighbouring diagonals, Applied Mathematics and Computation 10 (009) Download Date 4/10/18 :13 PM

7 Orthogonal diagonalization 79 [6] C. D. Meyer, Matrix Analysis and Applied Linear Algebra, SIAM, 000. [7] P. Lancaster, M. Tismenetsky, The Theory of Matrices, Academic Press, [8] J. Gutiérrez-Gutiérrez, Powers of complex persymmetric or skew-persymmetric anti-tridiagonal matrices with constant antidiagonals, Applied Mathematics and Computation 17 (011) [9] J. Lita da Silva, Integer powers of anti-tridiagonal matrices of the form antitridiag n (a, c, a), a, c C, International Journal of Computer Mathematics (015) DOI: / [10] J. Gutiérrez-Gutiérrez, Powers of tridiagonal matrices with constant diagonals, Applied Mathematics and Computation 06 (008) [11] T. M. Apostol, Calculus, Vol. 1, John Wiley & Sons, Download Date 4/10/18 :13 PM