Partial convergence and sign convergence of matrix powers via eigen c-ads.
Partial sum of eigenvalues of random graphs
Let be a graph on vertices and let be the eigenvalues of its adjacency matrix. For random graphs we investigate the sum of eigenvalues , for , and show that a typical graph has , where is the number of edges of . We also show bounds for the sum of eigenvalues within a given range in terms of the number of edges. The approach for the proofs was first used in Rocha (2020) to bound the partial sum of eigenvalues of the Laplacian matrix.
Paths of matrices with the strong Perron-Frobenius property converging to a given matrix with the Perron-Frobenius property.
Patterns with several multiple eigenvalues
Identified are certain special periodic diagonal matrices that have a predictable number of paired eigenvalues. Since certain symmetric Toeplitz matrices are special cases, those that have several multiple 5 eigenvalues are also investigated further. This work generalizes earlier work on response matrices from circularly symmetric models.
Pentadiagonal Companion Matrices
The class of sparse companion matrices was recently characterized in terms of unit Hessenberg matrices. We determine which sparse companion matrices have the lowest bandwidth, that is, we characterize which sparse companion matrices are permutationally similar to a pentadiagonal matrix and describe how to find the permutation involved. In the process, we determine which of the Fiedler companion matrices are permutationally similar to a pentadiagonal matrix. We also describe how to find a Fiedler...
Periodicity in quasipolynomial convolution.
Permanental associated matrices and Schneider's theorem.
Perron-Frobenius and Krein-Rutman theorems for tangentially positive operators
We study several aspects of a generalized Perron-Frobenius and Krein-Rutman theorems concerning spectral properties of a (possibly unbounded) linear operator on a cone in a Banach space. The operator is subject to the so-called tangency or weak range assumptions implying the resolvent invariance of the cone. The further assumptions rely on relations between the spectral and essential spectral bounds of the operator. In general we do not assume that the cone induces the Banach lattice structure into...
Perturbation of purely imaginary eigenvalues of Hamiltonian matrices under structured perturbations.
Perturbation theorems for the matrix eigenvalue problem
Perturbed spectra of defective matrices.
Perturbing non-real eigenvalues of nonnegative real matrices.
Poisson convergence for the largest eigenvalues of heavy tailed random matrices
We study the statistics of the largest eigenvalues of real symmetric and sample covariance matrices when the entries are heavy tailed. Extending the result obtained by Soshnikov in (Electron. Commun. Probab.9 (2004) 82–91), we prove that, in the absence of the fourth moment, the asymptotic behavior of the top eigenvalues is determined by the behavior of the largest entries of the matrix.
Polynomial numerical hulls of order 3.
Positive eigenvalues and two-letter generalized words.
Positive entries of stable matrices.
Principal eigenvectors of irregular graphs.
Product of operators and numerical range preserving maps
Let V be the C*-algebra B(H) of bounded linear operators acting on the Hilbert space H, or the Jordan algebra S(H) of self-adjoint operators in B(H). For a fixed sequence (i₁, ..., iₘ) with i₁, ..., iₘ ∈ 1, ..., k, define a product of by . This includes the usual product and the Jordan triple product A*B = ABA as special cases. Denote the numerical range of A ∈ V by W(A) = (Ax,x): x ∈ H, (x,x) = 1. If there is a unitary operator U and a scalar μ satisfying such that ϕ: V → V has the form A...
Pseudo linear transformations and evaluation in Ore extensions.
Pseudospectra and matrix behaviour
We study the extent to which the pseudospectra of a matrix determine other aspects of its behaviour, such as the growth of its powers and its unitary equivalence class.