ILAS2016 — 11–15 July 2016 — KU Leuven, Belgium

20th Conference of the International Linear Algebra Society (ILAS)

20th ILAS Conference

Contributed talks on Eigenvalue Algorithms and Pseudospectra

Tue 14:00–14:30, Room AV 03.12, Chair: Ding Lu
Singular points of the boundary of some pseudospectra
Gorka Armentia (Universidad Pública de Navarra)
Joint work with Juan-Miguel Gracia (Departamento de Matemática Aplicada y EIO, Universidad del País Vasco UPV/EHU); Francisco-Enrique Velasco (Departamento de Matemática Aplicada y EIO, Universidad del País Vasco UPV/EHU)

Let $A\in \mathbb{C}^{n\times n}$ and $\varepsilon >0$. Let us set $M(x,y)=(x+y \mathrm{i})I_n-A$ and bear in mind that $^*$ stands for the Hermitian transpose of a given matrix. For each $\varepsilon$, we denote by

\[ f_{\varepsilon}(x,y)=\det(\varepsilon I_n -M(x,y)^*M(x,y)) \]

the algebraic real curve that contains the boundary of the strict $\varepsilon$-pseudospectrum of $A$. Our goal is to determine and classify the singular points of this boundary. For the latter one, we use the Puiseux series; more precisely, for such a singular point there is an invariant under affine transformations, namely, the number of different branches that passes through this point and characterizes it. Until now, the only singular points we have found are tacnodes, acnodes, isolated points and cuspidal points. We provide some examples that illustrate these different cases.

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