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The classification of edges and the change in multiplicity of an eigenvalue of a real symmetric matrix resulting from the change in an edge value

Kenji Toyonaga, Charles R. Johnson (2017)

Special Matrices

We take as given a real symmetric matrix A, whose graph is a tree T, and the eigenvalues of A, with their multiplicities. Each edge of T may then be classified in one of four categories, based upon the change in multiplicity of a particular eigenvalue, when the edge is removed (i.e. the corresponding entry of A is replaced by 0).We show a necessary and suficient condition for each possible classification of an edge. A special relationship is observed among 2-Parter edges, Parter edges and singly...

The Collatz-Wielandt quotient for pairs of nonnegative operators

Shmuel Friedland (2020)

Applications of Mathematics

In this paper we consider two versions of the Collatz-Wielandt quotient for a pair of nonnegative operators A , B that map a given pointed generating cone in the first space into a given pointed generating cone in the second space. If the two spaces and two cones are identical, and B is the identity operator, then one version of this quotient is the spectral radius of A . In some applications, as commodity pricing, power control in wireless networks and quantum information theory, one needs to deal with...

The maximum multiplicity and the two largest multiplicities of eigenvalues in a Hermitian matrix whose graph is a tree

Rosário Fernandes (2015)

Special Matrices

The maximum multiplicity of an eigenvalue in a matrix whose graph is a tree, M1, was understood fully (froma combinatorial perspective) by C.R. Johnson, A. Leal-Duarte (Linear Algebra and Multilinear Algebra 46 (1999) 139-144). Among the possible multiplicity lists for the eigenvalues of Hermitian matrices whose graph is a tree, we focus upon M2, the maximum value of the sum of the two largest multiplicities when the largest multiplicity is M1. Upper and lower bounds are given for M2. Using a combinatorial...

Theorems of the alternative for cones and Lyapunov regularity of matrices

Bryan Cain, Daniel Hershkowitz, Hans Schneider (1997)

Czechoslovak Mathematical Journal

Standard facts about separating linear functionals will be used to determine how two cones C and D and their duals C * and D * may overlap. When T V → W is linear and K ⊂ V and D ⊂ W are cones, these results will be applied to C = T ( K ) and D , giving a unified treatment of several theorems of the alternate which explain when C contains an interior point of D . The case when V = W is the space H of n × n Hermitian matrices, D is the n × n positive semidefinite matrices, and T ( X ) = A X + X * A yields new and known results about the existence of block diagonal...

Tilings associated with non-Pisot matrices

Maki Furukado, Shunji Ito, E. Arthur Robinson (2006)

Annales de l’institut Fourier

Suppose A ∈ G l d ( ℤ ) has a 2-dimensional expanding subspace E u , satisfies a regularity condition, called “good star”, and has A * ≥ 0 , where A * is an oriented compound of A . A morphism θ of the free group on { 1 , 2 , ⋯ , d } is called a non-abelianization of A if it has structure matrix A . We show that there is a tiling substitution Θ whose “boundary substitution” θ = ∂ Θ is a non-abelianization of A . Such a tiling substitution Θ leads to a self-affine tiling of E u ∼ ℝ 2 with A u : = A | E u ∈ G L 2 ( ℝ ) as its expansion. In the last section we find conditions on A so...

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