Double Eigenvalue Research Articles

Banach Space Bifurcation Theory

We consider the bifurcation problem for the nonlinear operator equation $x = \lambda Lx + T(\lambda ,x,y)$ in a real Banach space $X$. Here ${\lambda _0}$ is an eigenvalue of the bounded linear operator $L,X = N(I - {\lambda _0}L) \oplus R(I - {\lambda _0}L),T \in {C^1}$ and $T$ is of higher order in $x$. New techniques are developed to simplify the solution of the bifurcation problem. When ${\lambda _0}$ is a simple eigenvalue, ${\lambda _0}$ is shown to be a bifurcation point of the homogeneous equation (i.e. $y \equiv 0$) with respect to 0. All solutions near $({\lambda _0},0)$ are shown to be of the form $(\lambda (\epsilon ),x(\epsilon )),0 \leqslant |\epsilon | < {\epsilon _0},\lambda (\epsilon )$ and $x(\epsilon )$ are continuous and $\lambda (\epsilon )$ and $x(\epsilon )$ are in ${C^n}$ or real analytic as $T$ is in ${C^{n + 1}}$ or is real analytic. When $T$ is real analytic and $\lambda (\epsilon ){\lambda _0}$ then there are at most two solution branches, and each branch is an analytic function of $\lambda$ for $\lambda \ne {\lambda _0}$. If $T$ is odd and analytic, for each $\lambda \in ({\lambda _0} - \delta ,{\lambda _0})$ (or $\lambda \in ({\lambda _0},{\lambda _0} + \delta )$) there exist two nontrivial solutions near 0 and there are no solutions near 0 for $\lambda \in ({\lambda _0},{\lambda _0} + \delta )$ (or $\lambda \in ({\lambda _0} - \delta ,{\lambda _0})$). We then demonstrate that in each sufficiently small neighborhood of a solution of the homogeneous bifurcation problem there are solutions of the nonhomogeneous equation (i.e. $y \not \equiv 0$) depending continuously on a real parameter and on $y$. If ${\lambda _0}$ is an eigenvalue of odd multiplicity we prove it is a point of bifurcation of the homogeneous equation. With a strong restriction on the projection of $T$ onto the null space of $I - {\lambda _0}L$ we show ${\lambda _0}$ is a bifurcation point of the homogeneous equation when ${\lambda _0}$ is a double eigenvalue. Counterexamples to some of our results are given when the hypotheses are weakened.

Banach space bifurcation theory

We consider the bifurcation problem for the nonlinear operator equation x = λ L x + T ( λ , x , y ) x = \lambda Lx + T(\lambda ,x,y) in a real Banach space X X . Here λ 0 {\lambda _0} is an eigenvalue of the bounded linear operator L , X = N ( I − λ 0 L ) ⊕ R ( I − λ 0 L ) , T ∈ C 1 L,X = N(I - {\lambda _0}L) \oplus R(I - {\lambda _0}L),T \in {C^1} and T T is of higher order in x x . New techniques are developed to simplify the solution of the bifurcation problem. When λ 0 {\lambda _0} is a simple eigenvalue, λ 0 {\lambda _0} is shown to be a bifurcation point of the homogeneous equation (i.e. y ≡ 0 y \equiv 0 ) with respect to 0. All solutions near ( λ 0 , 0 ) ({\lambda _0},0) are shown to be of the form ( λ ( ϵ ) , x ( ϵ ) ) , 0 ⩽ | ϵ | &gt; ϵ 0 , λ ( ϵ ) (\lambda (\epsilon ),x(\epsilon )),0 \leqslant |\epsilon | &gt; {\epsilon _0},\lambda (\epsilon ) and x ( ϵ ) x(\epsilon ) are continuous and λ ( ϵ ) \lambda (\epsilon ) and x ( ϵ ) x(\epsilon ) are in C n {C^n} or real analytic as T T is in C n + 1 {C^{n + 1}} or is real analytic. When T T is real analytic and λ ( ϵ ) λ 0 \lambda (\epsilon ){\lambda _0} then there are at most two solution branches, and each branch is an analytic function of λ \lambda for λ ≠ λ 0 \lambda \ne {\lambda _0} . If T T is odd and analytic, for each λ ∈ ( λ 0 − δ , λ 0 ) \lambda \in ({\lambda _0} - \delta ,{\lambda _0}) (or λ ∈ ( λ 0 , λ 0 + δ ) \lambda \in ({\lambda _0},{\lambda _0} + \delta ) ) there exist two nontrivial solutions near 0 and there are no solutions near 0 for λ ∈ ( λ 0 , λ 0 + δ ) \lambda \in ({\lambda _0},{\lambda _0} + \delta ) (or λ ∈ ( λ 0 − δ , λ 0 ) \lambda \in ({\lambda _0} - \delta ,{\lambda _0}) ). We then demonstrate that in each sufficiently small neighborhood of a solution of the homogeneous bifurcation problem there are solutions of the nonhomogeneous equation (i.e. y ≢ 0 y \not \equiv 0 ) depending continuously on a real parameter and on y y . If λ 0 {\lambda _0} is an eigenvalue of odd multiplicity we prove it is a point of bifurcation of the homogeneous equation. With a strong restriction on the projection of T T onto the null space of I − λ 0 L I - {\lambda _0}L we show λ 0 {\lambda _0} is a bifurcation point of the homogeneous equation when λ 0 {\lambda _0} is a double eigenvalue. Counterexamples to some of our results are given when the hypotheses are weakened.

Einstein Tensor and 3-Parameter Groups of Isometries with 2-Dimensional Orbits

The algebraic classification of the Weyl and Ricci tensors and the relation between them in a Riemann space with an isometry group possessing a nontrivial isotropy group are reviewed. All metrics with Minkowski signature, invariant under a 3-parameter isometry group with 2-dimensional orbits having nondegenerate metrics, are constructed from the group properties and are shown to have Ricci tensors with a double eigenvalue, and the orbits are shown to be surfaces of constant curvature. The null orbits are shown to have a triply degenerate eigenvalue of the Ricci tensor. The various additionally degenerate metrics are classified in further detail, extending the work of Plebański and Stachel.

Einstein Tensor and Spherical Symmetry

The classification of symmetric second-rank tensors in Minkowski space and its application to the Einstein tensor is reviewed. It is shown that, for spherically symmetric metrics, the Einstein tensor always has a spacelike double eigenvector; and the possible types of Einstein tensor that this degeneracy allows are discussed. A complete classification of all spherically symmetric metrics with two double eigenvalues is given. A study of the timelike eigencongruence, in the case when one timelike and two spacelike eigenvectors exist, is carried out. Canonical forms for the metric, the Einstein tensor, and the Weyl tensor (which is always of type D) are given for each of the various possible types.

The Simple Iterative Method Applied to a Matrix Which has a Dominant Double Real Eigenvalue

Suppose A is an n × n matrix with real elements, and that λ = λ1 is a dominant double real eigenvalue. We will assume for simplicity that none of the order eigenvalues are repeated. Then there are two possibilities (a)matrix is non-defective, and we can find two linearly independent eigenvectors corresponding to the double root λ1,(b)matrix is defective, and we have only one linearly independent eigenvectors corresponding to the double root λ1.