45,298 research outputs found
Symmetric Integer Matrices Having Integer Eigenvalues
We provide characterization of symmetric integer matrices for rank at most 2 that have integer spectrum and give some constructions for such matrices of rank 3. We also make some connection between Hanlon’s conjecture and integer eigenvalue problem
Symmetric Integer Matrices Having Integer Eigenvalues
We provide characterization of symmetric integer matrices for rank at most 2 that have integer spectrum and give some constructions for such matrices of rank 3. We also make some connection between Hanlon’s conjecture and integer eigenvalue problem
Symmetric Integer Matrices Having Integer Eigenvalues
We provide characterization of symmetric integer matrices for rank at most 2 that have integer spectrum and give some constructions for such matrices of rank 3. We also make some connection between Hanlon’s conjecture and integer eigenvalue problem
Integer symmetric matrices having all their eigenvalues in the interval [-2,2]
We completely describe all integer symmetric matrices that have all their
eigenvalues in the interval [-2,2]. Along the way we classify all signed
graphs, and then all charged signed graphs, having all their eigenvalues in
this same interval. We then classify subsets of the above for which the integer
symmetric matrices, signed graphs and charged signed graphs have all their
eigenvalues in the open interval (-2,2).Comment: 33 pages, 18 figure
Symmetrizable integer matrices having all their eigenvalues in the interval [-2,2]
The adjacency matrices of graphs form a special subset of the set of all
integer symmetric matrices. The description of which graphs have all their
eigenvalues in the interval [-2,2] (i.e., those having spectral radius at most
2) has been known for several decades. In 2007 we extended this classification
to arbitrary integer symmetric matrices.
In this paper we turn our attention to symmetrizable matrices. We classify
the connected nonsymmetric but symmetrizable matrices which have entries in
that are maximal with respect to having all their eigenvalues in [-2,2].
This includes a spectral characterisation of the affine and finite Dynkin
diagrams that are not simply laced (much as the graph result gives a spectral
characterisation of the simply laced ones).Comment: 20 pages, 11 figure
Small-span Hermitian matrices over quadratic integer rings
Building on the classification of all characteristic polynomials of integer
symmetric matrices having small span (span less than 4), we obtain a
classification of small-span polynomials that are the characteristic polynomial
of a Hermitian matrix over some quadratic integer ring. Taking quadratic
integer rings as our base, we obtain as characteristic polynomials some
low-degree small-span polynomials that are not the characteristic (or minimal)
polynomial of any integer symmetric matrix.Comment: 16 page
Cyclotomic matrices over real quadratic integer rings
We classify all cyclotomic matrices over real quadratic integer rings and we
show that this classification is the same as classifying cyclotomic matrices
over the compositum all real quadratic integer rings. Moreover, we enumerate a
related class of symmetric matrices; those matrices whose eigenvalues are
contained inside the interval [-2,2] but whose characteristic polynomials are
not in Z[x].Comment: 13 page
On equiangular lines in 17 dimensions and the characteristic polynomial of a Seidel matrix
For a positive integer, we find restrictions modulo on the
coefficients of the characteristic polynomial of a Seidel matrix
. We show that, for a Seidel matrix of order even (resp. odd), there are
at most (resp. ) possibilities for
the congruence class of modulo . As an application
of these results, we obtain an improvement to the upper bound for the number of
equiangular lines in , that is, we reduce the known upper bound
from to .Comment: 21 pages, fixed typo in Lemma 2.
Quantization of multidimensional cat maps
In this work we study cat maps with many degrees of freedom. Classical cat
maps are classified using the Cayley parametrization of symplectic matrices and
the closely associated center and chord generating functions. Particular
attention is dedicated to loxodromic behavior, which is a new feature of
two-dimensional maps. The maps are then quantized using a recently developed
Weyl representation on the torus and the general condition on the Floquet
angles is derived for a particular map to be quantizable. The semiclassical
approximation is exact, regardless of the dimensionality or of the nature of
the fixed points.Comment: 33 pages, latex, 6 figures, Submitted to Nonlinearit
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