论文信息 - On the possible multiplicities of the eigenvalues of a Hermitian matrix whose graph is a tree

On the possible multiplicities of the eigenvalues of a Hermitian matrix whose graph is a tree

We consider the general problem of determining which lists of multiplicities for the eigenvalues occur among Hermitian matrices the graph of whose off-diagonal entries is a given tree. Several restrictions are cited and a construction strategy is given. Together, these are sufficient to characterize all lists for each tree in two infinite classes: the double paths and generalized stars, and to tabulate all lists for trees on fewer than nine vertices. Such tables should be useful for formulating and dispelling general conjectures.

Charles R. Johnson | António Leal Duarte

[1] Warren E. Ferguson,et al. The construction of Jacobi and periodic Jacobi matrices with prescribed spectra , 1980 .

[2] I. Olkin,et al. Inequalities: Theory of Majorization and Its Applications , 1980 .

[3] P. Nylen,et al. Minimum-rank matrices with prescribed graph , 1996 .

[4] S. Parter. On the Eigenvalues and Eigenvectors of a Class of Matrices , 1960 .

[5] Charles R. Johnson,et al. Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[6] John S. Maybee,et al. Mechanical vibration trees , 1974 .

[7] Gerry Wiener. Spectral multiplicity and splitting results for a class of qualitative matrices , 1984 .

[8] Moody T. Chu,et al. Inverse Eigenvalue Problems , 1998, SIAM Rev..

[9] Charles R. Johnson,et al. The maximum multiplicity of an eigenvalue in a matrix whose graph is a tree , 1999 .

[10] Charles R. Johnson,et al. ON THE MINIMUM NUMBER OF DISTINCT EIGENVALUES FOR A SYMMETRIC MATRIX WHOSE GRAPH IS A GIVEN TREE , 2002 .