Eigenvector dynamics: General theory and some applications

Eigenvector dynamics: General theory and some applications.

We propose a general framework to study the stability of the subspace spanned by P consecutive eigenvectors of a generic symmetric matrix H0 when a small perturbation is added. This problem is relevant in various contexts, including quantum dissipation (H0 is then the Hamiltonian) and financial risk control (in which case H0 is the assets' return covariance matrix). We argue that the problem ca...

Some Applications of Homogeneous Dynamics to Number Theory

(a linear unimodular change of variables). Then the problem reduces to studying values of the standard form Sm,n of signature (m,n) applied to the collection of vectors of the form {gx | x ∈ Z}. And the dynamical approach consists of studying the action of the stabilizer of the form Sm,n on such collections. Problem 2. Given m vectors y1, . . . ,ym ∈ R n (viewed as linear forms x 7→ yi ·x, x ∈ ...

On Some Eigenvector-Eigenvalue Relations

This paper generalizes the well-known identity which relates the last components of the eigenvectors of a symmetric matrix A to the eigenvalues of A and of the matrix An−1, obtained by deleting the last row and column of A. The generalizations relate to matrices and to Sturm–Liouville equations.

Generalized Eigenvector Problem and Its Applications

A General Fredholm Theory and Applications

The theory described here results from an attempt to find a general abstract framework in which various theories, like Gromov-Witten Theory (GW), Floer Theory (FT), Contact Homology (CH) and more generally Symplectic Field Theory (SFT) can be understood from a general point of view. Let us describe the general landscape in a somewhat oversimplified form. The common feature (with the exception o...