Generalized Eigenspaces and Pseudospectra of Nonnormal and Defective Matrix-Valued Dynamical Systems

We consider nonnormal matrix-valued dynamical systems with discrete time. For an eigenvalue of matrix, the number of times it appears as a root of the characteristic polynomial is called the algebraic multiplicity. On the other hand, the geometric multiplicity is the dimension of the linear space of eigenvectors associated with that eigenvalue. If the former exceeds the latter, then the eigenvalue is said to be defective and the matrix becomes nondiagonalizable by any similarity transformation. The discrete-time of our dynamics is identified with the geometric multiplicity of the zero eigenvalue $\lambda_0=0$. Its algebraic multiplicity takes about half of the matrix size at $t=1$ and increases stepwise in time, which keeps excess to the geometric multiplicity until their coincidence at the final time. Our model exhibits relaxation processes from far-from-normal to near-normal matrices, in which the defectivity of $\lambda_0$ is recovering in time. We show that such processes are realized as size reductions of pseudospectrum including $\lambda_0$. Here the pseudospectra are the domains on the complex plane which are not necessarily exact spectra but in which the resolvent of matrix takes extremely large values. The defective eigenvalue $\lambda_0$ is sensitive to perturbation and the eigenvalues of the perturbed systems are distributed densely in the pseudospectrum including $\lambda_0$. By constructing generalized eigenspace for $\lambda_0$, we give the Jordan block decomposition for the resolvent of matrix and characterize the pseudospectrum dynamics. Numerical study of the systems perturbed by Gaussian random matrices supports the validity of the present analysis.