Perturbation Theory for Eigenvalues and Curves Inside the Numerical Range

Let A be an n × n n × n generic and irreducible matrix. We consider power series expansions for eigenvectors corresponding to the real part of e-iθA e - i θ A and use them to analyze and approximate the components of the Kippenhahn algebraic curve associated with the numerical range of A . We obtain information about the local behavior of the curve. As a result we obtain new equations for the series expansions, new conditions under which these components contain no line segments as well as conditions under which a component degenerates to a point. Applying these results to work of Mirman on compressions of unitary matrices, we show that in Mirman’s setting the components are locally asymptotically quadratic at all but finitely many points.