Numerical linear algebra and solvability of partial differential equations (Q1849424)

The goal of this paper is to describe a connection between the following seeming unrelated issues: a) Let \(A\in M_n(\mathbb{C})\) be an \(n\)-matrix with complex entries. Its eigenvalues, \(\lambda_1,\dots,\lambda_n\), which are the solutions of \(\text{det}(A-\lambda)=0\), are well-defined mathematical objects. Their numerical computation is a delicate problem which in the case when \(A\) is not normal \((AA^*\neq A^*A)\) may be very unstable. b) Let \(V\) be a nonvanishing vector field, \(V=\sum^3_{j=1}a_j(x)\partial_{x_j}\). Can the equation \(Vu=f\), \(f\in C^\infty (\mathbb{R}^3)\), be locally solved somewhere? That is, does there exist \(\Omega\subset \mathbb{R}^3\), open, and \(u\in C^1(\Omega)\) such that \(Vu=f\) in \(\Omega\). The author shows that some of the difficulties in finding eigenvalues for highly non-selfadjoint problems result from the phenomenon which also cause lack of solvability of most partial differential operators with complex coefficients.