Eigenvalues of matrices with tree graphs

AbstractWhen the undirected graph of a real square matrix is a tree of forest, we establish finitely computable tests yielding information about the magnitudes and multiplicities to the eigenvalues of the matrix. Applying these tests to system designs expressed as signed directed graphs can be sufficient to guarantee controllability of the associated linear dynamical systems

Some remarks on matrix stability with application to Turing instability

AbstractAn example of a 4×4 matrix is given that provides a counterexample to a result on Turing (diffusion-driven) instability and also answers negatively a conjecture on strong stability. Such instability is shown to arise from nonreal eigenvalues. The relevance and the connection of our example to classes of matrix stability known in the literature are discussed

Z-Pencils

The matrix pencil (A,B) = {tB-A | t \in C} is considered under the assumptions that A is entrywise nonnegative and B-A is a nonsingular M-matrix. As t varies in [0,1], the Z-matrices tB-A are partitioned into the sets L_s introduced by Fiedler and Markham. As no combinatorial structure of B is assumed here, this partition generalizes some of their work where B=I. Based on the union of the directed graphs of A and B, the combinatorial structure of nonnegative eigenvectors associated with the largest eigenvalue of (A,B) in [0,1) is considered.Comment: 8 pages, LaTe

Elementary bidiagonal factorizations

AbstractAn elementary bidiagonal (EB) matrix has every main diagonal entry equal to 1, and exactly one off-diagonal nonzero entry that is either on the sub- or super-diagonal. If matrix A can be written as a product of EB matrices and at most one diagonal matrix, then this product is an EB factorization of A. Every matrix is shown to have an EB factorization, and this is related to LU factorization and Neville elimination. The minimum number of EB factors needed for various classes of n-by-n matrices is considered. Some exact values for low dimensions and some bounds for general n are proved; improved bounds are conjectured. Generic factorizations that correspond to different orderings of the EB factors are briefly considered

Inertially arbitrary sign patterns with no nilpotent realization

AbstractAn n by n sign pattern S is inertially arbitrary if each ordered triple (n1,n2,n3) of nonnegative integers with n1+n2+n3=n is the inertia of some real matrix in Q(S), the sign pattern class of S. If every real, monic polynomial of degree n having a positive coefficient of xn−2 is the characteristic polynomial of some matrix in Q(S), then it is shown that S is inertially arbitrary. A new family of irreducible sign patterns G2k+1(k⩾2) is presented and proved to be inertially arbitrary, but not potentially nilpotent (and thus not spectrally arbitrary). The well-known Nilpotent-Jacobian method cannot be used to prove that G2k+1 is inertially arbitrary, since G2k+1 has no nilpotent realization. In order to prove that Q(G2k+1) allows each inertia with n3⩾1, a realization of G2k+1 with only zero eigenvalues except for a conjugate pair of pure imaginary eigenvalues is identified and used with the Implicit Function Theorem. Matrices in Q(G2k+1) with inertias having n3=0 are constructed by a recursive procedure from those of lower order. Some properties of the coefficients of the characteristic polynomial of an arbitrary matrix having certain fixed inertias are derived, and are used to show that G5 and G7 are minimal inertially arbitrary sign patterns

Spectra and inverse sign patterns of nearly sign-nonsingular matrices

AbstractA nearly sign-nonsingular (NSNS) matrix is a real n × n matrix having at least two nonzero terms in the expansion of its determinant with precisely one of these terms having opposite sign to all the other terms. Using graph-theoretic techniques, we study the spectra of irreducible NSNS matrices in normal form. Specifically, we show that such a matrix can have at most one nonnegative eigenvalue, and can have no nonreal eigenvalue z in the sector {z: |arg z| ⩽ κ(n − 1)}. We also derive results concerning the sign pattern of inverses of these matrices

Effects of General Incidence and Polymer Joining on Nucleated Polymerization in a Model of Prion Proliferation

Two processes are incorporated into a new model for transmissible prion diseases. These are general incidence for the lengthening process of infectious polymers attaching to and converting noninfectious monomers, and the joining of two polymers to form one longer polymer. The model gives rise to a system of three ordinary differential equations, which is shown to exhibit threshold behavior dependent on the value of the parameter combination giving the basic reproduction number R0. For R00 \u3e1, the system is locally asymptotic to a positive disease equilibrium. The effect of both general incidence and joining is to decrease the equilibrium value of infectious polymers and to increase the equilibrium value of normal monomers. Since the onset of disease symptoms appears to be related to the number of infectious polymers, both processes may significantly inhibit the course of the disease. With general incidence, the equilibrium distribution of polymer lengths is obtained and shows a sharp decrease in comparison to the distribution resulting from mass action incidence. Qualitative global results on the disease free and disease equilibria are proved analytically. Numerical simulations using parameter values from experiments on mice (reported in the literature) provide quantitative demonstration of the effects of these two processes

Eigenvalue location for nonnegative and Z-matrices

AbstractLet Lk0 denote the class of n × n Z-matrices A = tl − B with B ⩾ 0 and ϱk(B) ⩽ t < ϱk + 1(B), where ϱk(B) denotes the maximum spectral radius of k × k principal submatrices of B. Bounds are determined on the number of eigenvalues with positive real parts for A ϵ Lk0, where k satisfies, ⌊n2⌋ ⩽ k ⩽ n − 1. For these classes, when k = n − 1 and n − 2, wedges are identified that contain only the unqiue negative eigenvalue of A. These results lead to new eigenvalue location regions for nonnegative matrices

Impact of travel between patches for spatial spread of disease

A multipatch model is proposed to study the impact of travel on the spatial spread of disease between patches with different level of disease prevalence. The basic reproduction number for the ith patch in isolation is obtained along with the basic reproduction number of the system of patches, R(0). Inequalities describing the relationship between these numbers are also given. For a two-patch model with one high prevalence patch and one low prevalence patch, results pertaining to the dependence of R(0) on the travel rates between the two patches are obtained. For parameter values relevant for influenza, these results show that, while banning travel of infectives from the low to the high prevalence patch always contributes to disease control, banning travel of symptomatic travelers only from the high to the low prevalence patch could adversely affect the containment of the outbreak under certain ranges of parameter values. Moreover, banning all travel of infected individuals from the high to the low prevalence patch could result in the low prevalence patch becoming diseasefree, while the high prevalence patch becomes even more disease-prevalent, with the resulting number of infectives in this patch alone exceeding the combined number of infectives in both patches without border control. Under the set of parameter values used, our results demonstrate that if border control is properly implemented, then it could contribute to stopping the spatial spread of disease between patches