Abstract
Considered are combinatorially symmetric matrices, whose graph is a given tree, in view of the fact recent analysis shows that the geometric multiplicity theory for the eigenvalues of such matrices closely parallels that for real symmetric (and complex Hermitian) matrices. In contrast to the real symmetric case, it is shown that (a) the smallest example (13 vertices) of a tree and multiplicity list (3, 3, 3, 1, 1, 1, 1) meeting standard necessary conditions that has no real symmetric realizations does have a diagonalizable realization and for arbitrary prescribed (real and multiple) eigenvalues, and (b) that all trees with diameter < 8 are geometrically di-minimal (i.e., have diagonalizable realizations with as few of distinct eigenvalues as the diameter). This re-raises natural questions about multiplicity lists that proved subtly false in the real symmetric case. What is their status in the geometric multiplicity list case?
References
[1] C.R. Johnson and A. Leal-Duarte. The maximum multiplicity of an eigenvalue in a matrix whose graph is a tree. Linear and Multilinear Algebra 46:139–144 (1999).10.1080/03081089908818608Search in Google Scholar
[2] A. Leal-Duarte and C.R. Johnson. On the minimum number of distinct eigenvalues for a symmetric matrix whose graph is a given tree. Mathematical Inequalities and Applications 5(2):175–180 (2002).10.7153/mia-05-19Search in Google Scholar
[3] C.R. Johnson and A. Leal-Duarte. On the possible multiplicities of the eigenvalues of an Hermitian matrix whose graph is a given tree. Linear Algebra and its Applications 348:7–21 (2002).10.1016/S0024-3795(01)00522-5Search in Google Scholar
[4] C.R. Johnson, A. Leal-Duarte, C.M. Saiago, B.D. Sutton, and A.J. Witt. On the relative position of multiple eigenvalues in the spectrum of an Hermitian matrix with a given graph. Linear Algebra and its Applications 363:147–159 (2003).10.1016/S0024-3795(01)00589-4Search in Google Scholar
[5] C.R. Johnson, A. Leal-Duarte, and C.M. Saiago. Inverse eigenvalue problems and lists of multiplicities of eigenvalues for matrices whose graph is a tree: the case of generalized stars and double generalized stars. Linear Algebra and its Applications 373:311–330 (2003).10.1016/S0024-3795(03)00582-2Search in Google Scholar
[6] C.R. Johnson and C.M. Saiago. Branch duplication for the construction of multiple eigenvalues in an Hermitian matrix whose graph is a tree. Linear and Multilinear Algebra 56(4):357–380 (2008).10.1080/03081080600597668Search in Google Scholar
[7] C.R. Johnson, B.D. Sutton, and A. Witt. Implicit construction of multiple eigenvalues for trees. Linear and Multilinear Algebra 57(4):409–420 (2009).10.1080/03081080701852756Search in Google Scholar
[8] C.R. Johnson, J. Nuckols, and C. Spicer. The implicit construction of multiplicity lists for classes of trees and verification of some conjectures. Linear Algebra and its Applications 438(5):1990–2003 (2013).10.1016/j.laa.2012.11.010Search in Google Scholar
[9] C.R. Johnson, A.A. Li, and A.J. Walker. Ordered multiplicity lists for eigenvalues of symmetric matrices whose graph is a linear tree. Discrete Mathematics 333:39–55 (2014).10.1016/j.disc.2014.04.030Search in Google Scholar
[10] C.R. Johnson and C.M. Saiago. Diameter minimal trees. Linear and Multilinear Algebra 64(3):557–571 (2016).10.1080/03081087.2015.1057097Search in Google Scholar
[11] S.P. Buckley, J.G. Corliss, C.R. Johnson, C.A. Lombardía, and C.M. Saiago. Questions, conjectures, and data about multiplicity lists for trees. Linear Algebra and its Applications, 511:72–109 (2016).10.1016/j.laa.2016.08.002Search in Google Scholar
[12] C.R. Johnson and Y. Zhang. Multiplicity lists for symmetric matrices whose graphs have few missing edges. Linear Algebra and its Applications, 540:221–233 (2018).10.1016/j.laa.2017.11.032Search in Google Scholar
[13] C.R. Johnson and C.M. Saiago. Eigenvalues, Multiplicities and Graphs. Cambridge Tracts in Mathematics, Cambridge University Press, 2018.10.1017/9781316155158Search in Google Scholar
[14] C.R. Johnson and C.M. Saiago. Geometric Parter-Wiener, etc. Theory. Linear Algebra and its Applications, 537:332–347 (2018).10.1016/j.laa.2017.09.035Search in Google Scholar
[15] C.R. Johnson, A. Leal-Duarte, and C.M. Saiago. The minimum number of eigenvalues of multiplicity one in a diagonalizable matrix, over a field, whose graph is a tree. Linear Algebra and its Applications 559:1–10 (2018).10.1016/j.laa.2018.08.033Search in Google Scholar
[16] C.R. Johnson, C. Jordan-Squire, and D.A. Sher. Eigenvalue assignments and the two largest multiplicities in a Hermitian matrix whose graph is a tree. Discrete Applied Mathematics 158(6):681–691 (2010).10.1016/j.dam.2009.11.009Search in Google Scholar
[17] C.R. Johnson, J. Lettie, S. Mack-Crane, and A. Szabelska. Branch duplication in trees: Uniqueness of seed and enumeration of seeds. in manuscriptSearch in Google Scholar
[18] C.R. Johnson, A. Leal-Duarte, and C.M. Saiago. The Parter-Wiener theorem: refinement and generalization. SIAM Journal on Matrix Analysis and Applications 25(2):352–361 (2003).10.1137/S0895479801393320Search in Google Scholar
[19] F. Barioli and S.M. Fallat. On two conjectures regarding an inverse eigenvalue problem for acyclic symmetric matrices. Electronic Journal of Linear Algebra 11:41–50 (2004).10.13001/1081-3810.1120Search in Google Scholar
[20] R. Horn and C.R. Johnson. Matrix Analysis. Cambridge University Press, New York, 2nd Edition, 2013.Search in Google Scholar
[21] S. Parter. On the eigenvalues and eigenvectors of a class of matrices. Journal of the Society for Industrial and Applied Mathematics 8:376–388 (1960).10.1137/0108024Search in Google Scholar
[22] G. Wiener. Spectral multiplicity and splitting results for a class of qualitative matrices. Linear Algebra and its Applications 61:15–29 (1984).10.1016/0024-3795(84)90019-3Search in Google Scholar
© 2019 Carlos M. Saiago, published by De Gruyter
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