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 $\Z$ 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 $e$ a positive integer, we find restrictions modulo $2^e$ on the coefficients of the characteristic polynomial $\chi_S(x)$ of a Seidel matrix $S$. We show that, for a Seidel matrix of order $n$ even (resp. odd), there are at most $2^{\binom{e-2}{2}}$ (resp. $2^{\binom{e-2}{2}+1}$) possibilities for the congruence class of $\chi_S(x)$ modulo $2^e\mathbb Z[x]$. As an application of these results, we obtain an improvement to the upper bound for the number of equiangular lines in $\mathbb R^{17}$, that is, we reduce the known upper bound from $50$ to $49$.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