Perturbation Theory for Eigenvalues and Curves Inside the Numerical Range

Abstract Let A be an $$n \times n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>×</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:math> generic and irreducible matrix. We consider power series expansions for eigenvectors corresponding to the real part of $$e^{-i\theta }A$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>e</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mi>i</mml:mi> <mml:mi>θ</mml:mi> </mml:mrow> </mml:msup> <mml:mi>A</mml:mi> </mml:mrow> </mml:math> 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.