The paper presents a new general theory of interaction of eigenvalues of matrix operators depending on parameters. Both complex and real eigenvalues are considered. Strong and weak interactions are distinguished, and their geometric interpretation on the complex plane is given. Mechanical examples are presented and discussed in detail. r 2003 Elsevier Ltd. All rights reserved.

practically encountered complex eigenvalues sD of the matrix product D D g T f where Dg and Df are approximations to the connection matrix D on the signal and constraint sides of the neural net respectively. A stability criterion in the form of an analytic expression is derived thus generalizing the results obtained by Yan [8] for the special case of purely real eigenvalues.

A bstract As a generalization of entanglement entropy, pseudo entropy is not always real. The real-valued has promising applications in holography and quantum phase transition. We apply the notion pseudo-Hermiticity to formulate reality condition entropy. find general form transition matrix for which eigenvalues reduced possess real or complex pairs eigenvalues. Further, we class matrices (Rény...

We use the idea of a Wigner surmise to compute approximate distributions of the first eigenvalue in chiral random matrix theory, for both real and complex eigenvalues. Testing against known results for zero and maximal non-Hermiticity in the microscopic large- N limit, we find an excellent agreement valid for a small number of exact zero eigenvalues. Compact expressions are derived for real eig...

The eigenvalues of a matrix A are the zeros of its characteristic polynomial fiX) = dtt[A XI]. With Hyman's method of determinant evaluation, a new homotopy continuation method, homotopy-determinant method, is developed in this paper for finding all eigenvalues of a real upper Hessenberg matrix. In contrast to other homotopy continuation methods, the homotopy-determinant method calculates eigen...

The inverse problem of constructing a symmetric Toeplitz matrix with prescribed eigenvalues has been a challenge both theoretically and computationally in the literature. It is now known in theory that symmetric Toeplitz matrices can have arbitrary real spectra. This paper addresses a similar problem — Can the three largest eigenvalues of symmetric pentadiagonal Toeplitz matrices be arbitrary? ...

We present a direct method for reordering eigenvalues in the generalized periodic real Schur form of a regular K-cyclic matrix pair sequence (Ak, Ek). Following and generalizing existing approaches, reordering consists of consecutively computing the solution to an associated Sylvester-like equation and constructingK pairs of orthogonal matrices. These pairs define an orthogonal K-cyclic equival...

According to Kippenhahn’s classification, numerical ranges W (A) of unitarily irreducible 3 × 3 matrices A come in three possible shapes, an elliptical disk being one of them. The known criterion for the ellipticity of W (A) consists of several equations, involving the eigenvalues of A. It is shown herein that the set of 3× 3 matrices satisfying these conditions is nowhere dense, i.e., one of t...

Let A(G) denote the adjacency matrix of the graph G. The polynomial pA(G)(x) is usually referred to as the characteristic polynomial of G. For convenience, we use p(G, x) to denote pA(G)(x). The spectrum of a graph G is the set of eigenvalues of A(G) together with their multiplicities. Since A (short for A(G)) is a real symmetric matrix, basic linear algebra tells us a few thing about A and its...

Let t 7→ A(t) for t ∈ T be a C -mapping with values unbounded operators with compact resolvents and common domain of definition which are self-adjoint or normal. Here C stands for C (real analytic), a quasianalytic or non-quasianalytic Denjoy-Carleman class, C∞, or a Hölder continuity class C. The parameter domain T is either R or R or an infinite dimensional convenient vector space. We prove a...