is the jacobian matrix of F . If this matix has n distinct eigenvalues λk (real or complex), the system can be diagonalized, and the solutions can be expressed as combinations of exponentials of the form ek. Therefore, x − x∗ → 0 if all the eigenvalues have a negative real part. If one or more eigenvalues have positive real part, |x− x∗| → ∞. Therefore, • The fixed point x∗ is stable if all the...

The undirected power graph of a finite group $G$, $P(G)$, is a graph with the group elements of $G$ as vertices and two vertices are adjacent if and only if one of them is a power of the other. Let $A$ be an adjacency matrix of $P(G)$. An eigenvalue $lambda$ of $A$ is a main eigenvalue if the eigenspace $epsilon(lambda)$ has an eigenvector $X$ such that $X^{t}jjneq 0$, where $jj$ is the all-one...

In this manuscript, a numerical technique is presented for finding the eigenvalues of the regular Sturm-Liouville problems. The Chebyshev cardinal functions are used to approximate the eigenvalues of a regular Sturm-Liouville problem with Dirichlet boundary conditions. These functions defined by the Chebyshev function of the first kind. By using the operational matrix of derivative the problem ...

Stability analysis of spatially discretized approximations to invariant circle solutions to a noninvertible map is shown to produce seemingly spurious eigenvalues, found to exist for all truncation numbers. Numerical analysis of the spectral Galerkin projection procedure used to compute the solutions and the eigenvalues of the linearization shows that the set of discrete eigenvalues converges i...

Matrix nearness problems have been the focus of much research in linear algebra; see, e.g., [8,9,19,32]. In particular, characterizations of the algebraic variety of normal matrices and distance measures to this variety have received considerable attention; see [10,11,13,14,16,17,23,25–27]. Normal matrices are of interest because their eigenvalues are optimally conditioned and their singular va...

In this paper, we explore what happens when the same techniques are applied to the problem of estimating eigenvalues of the adjacency operator on finite graphs of bounded degree. In Theorem 7, we show how eigenvalues of the adjacency operator on a finite graph Γ may be bounded in terms of the biggest eigenvalues of the adjacency operator on “geodesic balls” in Γ. We find explicit bounds for the...

If the adjacency matrix contains degenerate eigenvalues, we must modify the approach using non-degenerate eigenvalues. We denote the eigenvalues as λki, where the index k runs over different eigenvalues and the index i runs over M associated eigenvectors of the same eigenvalue. Note that there is no unique way of choosing a basis for the eigenvectors of the unperturbed network since any linear ...

Most pattern recognition applications require the eigenvalues and eigenvectors of the covariance matrix. It is well known that when the number of training samples is small, the eigenvalues of the covariance matrix contains bias, and the bias degrades recognition performance. There are some methods which ignore the small eigenvalues, or acquire better estimates of the covariance matrix by correc...