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.
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Bini, D., Capovani, M.: Spectral and computational properties of band symmetric Toeplitz matrices. Linear Algebra Appl. 52(53), 99–126 (1983)
Bultheel, A., Van Barel, M.: Linear Algebra: Rational Approximation and Orthogonal Polynomials. Studies in Computational Mathematics, vol. 6. Elsevier Science B.V., Amsterdam (1997)
Bunch, J.R., Nielsen, C.P., Sorensen, D.C.: Rank-one modification of the symmetric eigenproblem. Numer. Math. 31, 31–48 (1978)
Datta, B.N., Johnson, C.R., Kaashoek, M.A., Plemmons, R.J., Sontag, E.D.: Linear Algebra in Signals, Systems, and Control. Society for Industrial and Applied Mathematics, Philadelphia (1988)
Fasino, D.: Spectral and structural properties of some pentadiagonal symmetric matrices. Calcolo 25(4), 301–310 (1988)
Ford, W.: Numerical Linear Algebra with Applications Using MATLAB. Academic Press, Cambridge (2015)
Gutiérrez-Gutiérrez, J.: Powers of complex persymmetric or skew-persymmetric anti-tridiagonal matrices with constant anti-diagonals. Appl. Math. Comput. 217, 6125–6132 (2011)
Gutiérrez-Gutiérrez, J.: Eigenvalue decomposition for persymmetric Hankel matrices with at most three non-zero anti-diagonals. Appl. Math. Comput. 234, 333–338 (2014)
Harville, D.A.: Matrix Algebra From a Statistician’s Perspective. Springer, New York (1997)
Honglin, W.: On computing of arbitrary positive powers for one type of anti-tridiagonal matrices of even order. Appl. Math. Comput. 217, 2750–2756 (2010)
Horn, R.A., Johnson, C.R.: Matrix Analysis, 2nd edn. Cambridge University Press, New York (2013)
Jain, P., Pachori, R.B.: An iterative approach for decomposition of multi-component non-stationary signals based on eigenvalue decomposition of the Hankel matrix. J. Frankl. Inst. 352(10), 4017–4044 (2015)
Kailath, T.: Linear Systems. Prentice-Hall, Englewood Cliffs (1980)
Lita da Silva, J.: On anti-pentadiagonal persymmetric Hankel matrices with perturbed corners. J. Comput. Math. Appl. 72, 415–426 (2016)
Miller, K.S.: On the inverse of the sum of matrices. Math. Mag. 54(2), 67–72 (1981)
Olshevsky, V., Stewart, M.: Stable factorization for Hankel and Hankel-like matrices. Numer. Linear Algebra Appl. 8, 401–434 (2001)
Perko, L.: Differential Equations and Dynamical Systems, 3rd edn. Springer, New York (2001)
Pissanetsky, S.: Sparse Matrix Technology. Academic Press, London (1984)
Rimas, J.: Integer powers of real odd order skew-persymmetric anti-tridiagonal matrices with constant anti-diagonals \(({\text{antitridiag}}_{n}(a, c,-a), a \in R \setminus \{0 \}, c \in R)\). Appl. Math. Comput. 219, 7075–7088 (2013)
Rimas, J.: Integer powers of real even order anti-tridiagonal Hankel matrices of the form \({\rm antitridiag}_{n}(a, c,-a)\). Appl. Math. Comput. 225, 204–215 (2013)
Williams, G.: Linear Algebra with Applications, 8th edn. Jones and Bartlett Learning, Burlington (2014)
Yin, Q.: On computing of arbitrary positive powers for anti-tridiagonal matrices of even order. Appl. Math. Comput. 203, 252–257 (2008)
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This work is a contribution to the Project UIDB/04035/2020, funded by FCT-Fundação para a Ciência e a Tecnologia, Portugal.
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Communicated by Fuzhen Zhang.
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Lita da Silva, J. Spectral properties of anti-heptadiagonal persymmetric Hankel matrices. Banach J. Math. Anal. 14, 1387–1420 (2020). https://doi.org/10.1007/s43037-020-00066-x
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DOI: https://doi.org/10.1007/s43037-020-00066-x