Abstract
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?
Highlights
In the last three decades, there has been considerable study of the possible multiplicity lists for the eigenvalues of real symmetric (Hermitian) matrices, whose graph is a given tree [1–12, etc.]
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 matrices
The maximum multiplicity is known [1], there is a close lower bound on the minimum number of distinct eigenvalues [2], and methods for getting all multiplicity lists for several classes of trees [3,4,5,6, 8, 9]
Summary
In the last three decades, there has been considerable study of the possible multiplicity lists for the eigenvalues of real symmetric (Hermitian) matrices, whose graph is a given tree (and more general graphs, as well) [1–12, etc.]. Certain of these conjectures have proven false in the general symmetric case (geometric/algebraic multiplicities in real symmetric matrices whose graph is a given tree), though the smallest counterexamples require large numbers of vertices (13 in one case and 16 in another). We explicitly show here that this assignment is geometrically realizable by a diagonalizable combinatorially symmetric matrix.
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