Inverse Eigenvalue Problems with Partial Eigen Data for Acyclic Matrices whose Graph is a Broom
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In this paper, we consider three inverse eigenvalue problems for a special type of acyclic matrices. The acyclic matrices considered in this paper are described by a graph called a broom on $n+m$ vertices, which is obtained by joining $m$ pendant edges to one of the terminal vertices of a path on $n$ vertices. The problems require the reconstruction of such a matrix from given partial eigen data. The eigen data for the first problem consists of the largest eigenvalue of each of the leading principal submatrices of the required matrix, while for the second problem it consists of an eigenvalue of each of its trailing principal submatrices. The third problem has an eigenvalue and a corresponding eigenvector of the required matrix as the eigen data. The method of solution involves the use of recurrence relations among the leading/trailing principal minors of $\lambda I-A$, where $A$ is the required matrix. We derive the necessary and sufficient conditions for the solutions of these problems. The constructive nature of the proofs also provides the algorithms for computing the required entries of the matrix. We also provide some numerical examples to show the applicability of our results.
Abstract
In this paper, we consider three inverse eigenvalue problems for a special type of acyclic matrices. The acyclic matrices considered in this paper are described by a graph called a broom on $n+m$ vertices, which is obtained by joining $m$ pendant edges to one of the terminal vertices of a path on $n$ vertices. The problems require the reconstruction of such a matrix from given partial eigen data. The eigen data for the first problem consists of the largest eigenvalue of each of the leading principal submatrices of the required matrix, while for the second problem it consists of an eigenvalue of each of its trailing principal submatrices. The third problem has an eigenvalue and a corresponding eigenvector of the required matrix as the eigen data. The method of solution involves the use of recurrence relations among the leading/trailing principal minors of $\lambda I-A$, where $A$ is the required matrix. We derive the necessary and sufficient conditions for the solutions of these problems. The constructive nature of the proofs also provides the algorithms for computing the required entries of the matrix. We also provide some numerical examples to show the applicability of our results.
참고문헌 (17)
- M. T. Chu, "Inverse eigenvalue problems", SIAM, 40, pp.1~39 (1998)
- A. L. Duarte, "Construction of acyclic matrices from spectral data", Linear Algebra Appl., 113, pp.173~182 (1989)
- A Leal-Duarte, "On the minimum number of distinct eigenvalues for a symmetric matrix whose graph is a given tree", Math. Inequal. Appl., 5, pp.175~180 (2002)
- S. Elhay, "On some eigenvectoreigenvalue relations", SIAM J. Matrix Anal. Appl, 20(3), pp.563~574 (1999)
- K. Ghanbari, "m-functions and inverse generalized eigenvalue problem", Inverse Problems, 17(2), pp.211~217 (2001)
- K. Ghanbari, "Generalized inverse eigenvalue problem with mixed eigen data", Linear Algebra Appl., 437, pp.2056~2063 (2012)
- G. M. L. Gladwell, "Inverse problems in vibration" (2004), Kluwer Academic Publishers
- L. Hogben, "Spectral graph theory and the inverse eigenvalue problem of a graph", Electron. J. Linear Algebra, 14, pp.12~31 (2005)
- R. A. Horn, "Matrix Analysis" (1986), Cambridge University Press
- C. R. Johnson, "The Parter Wiener theorem:refinement and generalization", SIAM J.Matrix Anal. Appl., 25, pp.352~361 (2003)
- K. H. Monfared, "Construction of matrices with a given graph and prescribed interlaced spectral data", Linear Algebra Appl., 438, pp.4348~4358 (2013)
- R. Nair, "Acyclic matrices with a small number of distinct eigenvalues", Linear Algebra and Appl., 438, pp.4075~4089 (2013)
- P. Nylen, "Inverse Eigenvalue Problems Associated With SpringMass Systems", Linear Algebra and Appl., 254, pp.409~425 (1997)
- J. Peng, "Two inverse eigenvalue problems for a special kind of matrices", Linear Algebra Appl., 416, pp.336~347 (2006)
- M. Sen, "Generalized inverse eigenvalue problem for matrices whose graph is a path", Linear Algebra Appl., 446, pp.224~236 (2014)
- D. Sharma, "Inverse Eigenvalue Problems for Two Special Acyclic Matrices", Mathematics, 4(12) (2016)
- Q. Su, "Inverse spectral problem for pseudo-Jacobi matrices with partial spectral data", J. Comput. Appl. Math., 297, pp.1~12 (2016)