The characteristic polynomial of some anti-tridiagonal 2-Hankel matrices of even order


  • João Lita da Silva Faculty of Sciences and Technology, NOVA University of Lisbon


Anti-tridiagonal 2-Hankel matrix, eigenvalue, eigenvector, Chebyshev polynomials


In this paper we derive the characteristic polynomial for a family of anti-tridiagonal 2-Hankel matrices of even order at the expense of Chebyshev polynomials giving also a representation of its eigenvectors. An orthogonal diagonalization for these type of matrices having null northeast-to-southwest diagonal is still provided using prescribed eigenvalues.


Akbulak, M., da Fonseca, C.M. & Yilmaz, F. (2013) The eigenvalues of a family of persymmetric anti-tridiagonal 2-Hankel matrices. Applied Mathematics and Computation, 225:352-357.

Alvarez-Nodarse, R., Petronilho, J. & Quintero, N.R. (2005) On some tridiagonal k-Toeplitz matrices: algebraic and analytical aspects. Applications. Journal of Computational and Applied Mathematics, 184:518-537.

Bultheel, A. & Van Barel, M. (1997) Linear algebra: rational approximation and orthogonal polynomials. Studies in Computational Mathematics 6, North-Holland, Amsterdam.

Chihara, T.S. (1978) An introduction to orthogonal polynomials. Gordon and Breach, New York.

da Fonseca, C.M. & Petronilho, J. (2001) Explicit inverses of some tridiagonal matrices. Linear Algebra and its Applications, 325:7-21.

da Fonseca, C.M. (2018) The eigenvalues of some anti-tridiagonal Hankel matrices. Kuwait Journal of Sciences, 45:1--6.

Gover, M.J.C. & Barnett, S. (1985) Inversion of Toeplitz matrices which are not strongly non-singular. IMA Journal of Numerical Analysis, 5:101-110.

Gover, M.J.C. & Barnett, S. (1985) Characterisation and properties of r-Toeplitz matrices. Journal of Mathematical Analysis and Applications, 123:297-305.

Gover, M.J.C. (1989) The determination of companion matrices characterizing Toeplitz and r-Toeplitz matrices. Linear Algebra and its Applications, 117:81-92.

Laub, A.J. (2005) Matrix analysis for scientists & engineers. SIAM, Philadelphia.

Mason, J.C. & Handscomb, D. (2003) Chebyshev polynomials. Chapman & Hall/CRC, Boca Raton.

Olshevsky, V. & Stewart, M. (2001) Stable factorization for Hankel and Hankel-like matrices. Numerical Linear Algebra with Applications, 8:401-434.

Wu, H. (2010) On computing of arbitrary positive powers for one type of anti-tridiagonal matrices of even order. Applied Mathematics and Computation, 217:2750-2756.