Abstract
Abstract Using the recent geometric Parter-Wiener, etc. theorem and related results, it is shown that much of the multiplicity theory developed for real symmetric matrices associated with paths and generalized stars remains valid for combinatorially symmetric matrices over a field. A characterization of generalized stars in the case of combinatorially symmetric matrices is given.
Highlights
The graph of the real symmetric matrix A = ∈ Mn(R) is the graph G on n vertices, . . . , n, with an edge {i, j} if and only if aij ≠
Using the recent geometric Parter-Wiener, etc. theorem and related results, it is shown that much of the multiplicity theory developed for real symmetric matrices associated with paths and generalized stars remains valid for combinatorially symmetric matrices over a eld
It has been shown that much of this multiplicity theory generalizes to geometric multiplicity for eigenvalues of combinatorially symmetric matrices (i.e., matrices A = with aij ≠ if and only if aji ≠ ) over a eld F
Summary
Theorem and related results, it is shown that much of the multiplicity theory developed for real symmetric matrices associated with paths and generalized stars remains valid for combinatorially symmetric matrices over a eld. E-mail: crjohn@wm.edu Hannah Lang: Department of Mathematics, Harvard University, Cambridge, MA, 02138, USA. Several other known results generalize to matrices in F(T) as well: the downer branch mechanism and that the maximum geometric multiplicity gM(T) ( rst discussed in [1]) of an eigenvalue of a matrix in F(T) is equal to the path cover number P(T), [4], [5].
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