On the inverse eigenproblem for symmetric and nonsymmetric arrowhead matrices

Resumen

We present a new construction of a symmetric arrow matrix from a particular spectral information: let λ1(n) be the minimal eigenvalue of the matrix and λj(j) ,j=1,2,...,n the maximales eigenvalues of all leading principal submatrices of the matrix. We use such a procedure the to construct a nonsymmetric arrow matrix from the same spectral information next to an eigenvector x(n)=(x1,x2, ,…,xn), so that (x(n),λn(n)) is an eigenpair of the matrix. Moreover our results generate an algorithmic procedure to compute a solution matrix.        

Biografía del autor/a

H. Pickmann, Universidad de Tarapacá.
Dept. de Matemática.
S. Arela, Universidad de Tarapacá.
Dept. de Matemática.
J. Egaña, Universidad Católica del Norte.
Dept. de Matemáticas.
D. Carrasco, Universidad del Bío-Bío.
Dept de Matemática.

Citas

M. Bixon and J. Jortner, “Intramolecular radiationless transitions”, The Journal of chemical physics, vol. 48, no. 2, pp. 715-726, 1968, doi: 10.1063/1.1668703.

M. Chu and G. Golub, Inverse eigenvalue problems: theory, algorithms, and applications. Oxford: Oxford University Press, 2005, doi: 10.1093/acprof:oso/9780198566649.001.0001.

E. Jessup, “A case against a divide and conquer approach to the nonsymmetric eigenvalue problema”, Applied numerical mathematics, vol. 12, no. 5, pp. 403-420, Jul. 1993, doi: 10.1016/0168-9274(93)90101-V.

V. Higgins and C. Johnson, “Inverse spectral problems for collections of leading principal submatrices of tridiagonal matrices”, Linear algebra and its applications, vol. 489, pp. 104-122, Jan. 2016, doi: 10.1016/j.laa.2015.10.004.

A. Nazari and Z. Beiranvand, “The inverse eigenvalue problem for symmetric quasi anti-bidiagonal matrices”, Applied mathematics and computation, vol. 217, no. 23, pp. 9526-9531, Aug. 2011, doi: 10.1016/j.amc.2011.03.031.

Z. Li, C. Bu and W. Hui, “Inverse eigenvalue problem for generalized arrow-like matrices”, Applied mathematics, vol. 2, no. 12, pp. 1443-1445, Dec. 2011, doi: 10.4236/am.2011.212204.

J. Peng, X. Hu and L. Zhang, “Two inverse eigenvalue problems for a special kind of matrices”, Linear algebra and its applications, vol. 416, no. 2-3, pp. 336-347, Jul. 2006, doi: 10.1016/j.laa.2005.11.017.

H. Pickmann, J. Egaña and R. Soto, “Extremal inverse eigenvalue problema for bordered diagonal matrices”, Linear algebra and its applications , vol. 427, no. 2-3, pp. 256-271, Dec. 2007, doi: 10.1016/j.laa.2007.07.020.

H. Pickmann, J. Egaña and R. Soto, “Extreme spectra realization by real symmetric tridiagonal and real symmetric arrow matrices”, Electronic journal of linear algebra, vol. 22, pp. 780-795, 2011, doi: 10.13001/1081-3810.1474.

H. Pickmann, S. Arela, J. Egaña and R. Soto, “Extreme spectra realization by real nonsymmetric tridiagonal and real nonsymmetric arrow matrices”, Mathematical problems in engineering, vol. 2019, Article ID 3459017, Mar. 2019, doi: 10.1155/2019/3459017.

H. Najafi, S. Edalatpanah and G. Gravvanis, “An efficient method for computing the inverse of arrowhead matrices”, Applied mathematics letters, vol. 33, pp. 1-5, Jul. 2014, doi: 10.1016/j.aml.2014.02.010.

L. Shen and B. Suter, “Bounds for eigenvalues of arrowhead matrices and their applications to hub matrices and wireless communications”, EURASIP Journal on advances in signal processing, vol. 2009, Article ID 379402, Dec. 2019, doi: 10.1155/2009/379402.

W. Wanicharpichat, “Explicit minimum polynomial, eigenvector, and inverse formula for nonsymmetric arrowhead matrix”, International journal of pure and applied mathematics, vol. 108, no. 4, pp. 967-984, 2016, doi: 10.12732/ijpam.v108i4.21.

Publicado
2019-10-22
Cómo citar
[1]
H. R. Pickmann Soto, S. Arela Pérez, J. C. Egaña Arancibia, y D. Carrasco Olivera, «On the inverse eigenproblem for symmetric and nonsymmetric arrowhead matrices», Proyecciones (Antofagasta, En línea), vol. 38, n.º 4, pp. 811-828, oct. 2019.
Sección
Artículos