The eigenvalues of some anti-tridiagonal Hankel matrices

Carlos Fonseca

Abstract


We determine the spectra of two families of anti-tridiagonal Hankel matrices of any order. The approach is much stronger and more concise than those particular cases appearing in the literature. At the same time, it simplifies significantly all the known results up to now.


Keywords


Anti-tridiagonal matrices; Hankel matrices; Chebyshev polynomials, eigenvalues

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References


Akbulak, M. , da Fonseca, C.M. & Ylmaz,

F. (2013). The eigenvalues of a family of persymmetric

anti-tridiagonal 2-Hankel matrices, Applied

Mathematics and Computation 225:352-357. DOI:

1016/j.amc.2013.09.014

da Fonseca, C.M. (2007). On the eigenvalues

of some tridiagonal matrices, Journal of Computational

and Applied Mathematics 200(1):283-286.

DOI: 10.1016/j.cam.2005.08.047

da Fonseca, C.M. & Petronilho, J. (2005).

Explicit inverse of a tridiagonal k-Toeplitz matrix,

Numerische Mathematik 100(3): 457-482. DOI:

1007/s00211-005-0596-3

da Fonseca, C.M. & Petronilho, J. (2001).

Explicit inverses of some tridiagonal matrices, Linear

Algebra and its Applications 325(1-3):7-21.

DOI: 10.1016/S0024-3795(00)00289-5

Gover, M.J.C. (1994). The eigenproblem

of a tridiagonal 2-Toeplitz matrix, Linear Algebra

and its Applications 197/198:63-78. DOI:

1016/0024-3795(94)90481-2

Gutierrez-Gutierrez, J. (2011). Powers

of complex persymmetric or skewpersymmetric

anti-tridiagonal matrices with

constant anti-diagonals, Applied Mathematics

and Computation 217(13):6125-6132. DOI:

1016/j.amc.2010.12.091

Gutierrez-Gutierrez, J. (2008). Powers

of real persymmetric anti-tridiagonal matrices

with constant anti-diagonals, Applied Mathematics

and Computation 206(2):919-924. DOI:

1016/j.amc.2008.10.003

Gutierrez-Gutierrez, J. & Zarraga-

Rodrguez, M. (2016). Orthogonal diagonalization

for complex skew-persymmetric antitridiagonal

Hankel matrices, Special Matrices 4:73-

DOI: 10.1515/spma-2016-0008

El-Mikkawy, M. & Rahmo, E-L. (2008).

A new recursive algorithm for inverting general

tridiagonal and anti-tridiagonal matrices, Applied

Mathematics and Computation 204(1):368-372.

DOI: 10.1016/j.amc.2008.06.053

Rimas, J. (2013). Integer powers of

real even order anti-tridiagonal Hankel matrices

of the form antitridiagn(a; c;􀀀a), Applied

Mathematics and Computation 225:204-215.

1016/j.amc.2013.09.029

Rimas, J. (2013). Integer powers of

real odd order skew-persymmetric antitridiagonal

matrices with constant anti-diagonals

(antitridiagn(a; c;􀀀a); a 2 R n f0g; c 2 R), Applied

Mathematics and Computation 219(12):7075-7088.

DOI: 10.1016/j.amc.2012.12.051

Rimas, J. (2012). Explicit expression for powers

of tridiagonal 2-Toeplitz matrix of odd order, Linear

Algebra and its Applications 436(9):3493-3506.

DOI: 10.1016/j.laa.2011.12.025

Rimas, J. (2009). On computing of arbitrary integer

powers of odd oirder anti-tridiagonal matrices

with zeros in main skew diagonal and elements

; 1; 1; : : : ; 1; 􀀀1;􀀀1;􀀀1; : : : ;􀀀1 in neighbouring diagonals,

Applied Mathematics and Computation

(1):64-71. DOI: 10.1016/j.amc.2008.11.001

Rimas, J. (2008). 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 in neighbouring diagonals,

Applied Mathematics and Computation

(2):754-763. DOI: /10.1016/j.amc.2008.07.021

Rimas, J. (2008). On computing of arbitrary positive

integer powers for one type of symmetric antitridiagonal

matrices of odd order, Applied Mathematics

and Computation 203(2):573-581. DOI:

1016/j.amc.2008.05.008

Rimas, J. (2008). On computing of arbitrary

integer powers for one type of symmetric antitridiagonal

matrices of even order, Applied Mathematics

and Computation 204(1):288-298. DOI:

1016/j.amc.2008.06.044

da Silva, J.L. (2015). Integer powers

of anti-tridiagonal matrices of the for

antitridiagn(a; c;􀀀a); a 2 R, Computers & Mathematics

with Applications 69:1313-1328. DOI:

1016/j.camwa.2015.03.016

Wang, H. (2014). Powers of complex persymmetric

antitridiagonal matrices with constant antidiagonals,

ISRN Computational Mathematics 2014:

Art. 451270. DOI: 10.1155/2014/451270

Wu, H. (2010). On computing of arbitrary

positive powers for one type of antitridiagonal

matrices of even order, Applied Mathematics

and Computation 217(6):2750-2756. DOI:

1016/j.amc.2010.08.010

Yin, Q. (2008). On computing of arbitrary

positive powers for anti-tridiagonal matrices of

even order, Applied Mathematics and Computation

(1):252-257. DOI: 10.1016/j.amc.2008.04.031


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