# NUMERICAL LINEAR ALGEBRA AND SOLVABILITY OF PDEs

## ABSTRACT

```
One of the goals of Numerical Linear Algebra is the accurate computation
of eigenvalues of matrices. This is quite successful in the symmetric case
but the situation is different for non-self adjoint matrices. These arise
in the discretisation of engineering or physical problems involving
friction or dissipation, and it is an important problem to compute their
eigenvalues.

It has been observed long time ago that the obstruction to accurately
computing eigenvalues of nonselfadjoint matrices is inherent in the
problem, and cannot be circumvented by using more powerful computers. The
basic idea is that algorithms for locating the eigenvalues may also find
some  `false eigenvalues.'

These false eigenvalues also explain one of the most surprising phenomena
in linear PDEs, namely the fact (discovered by Hans Lewy in 1957, in
Berkeley) that one cannot always locally solve the PDE \$Pu = f\$ . Local
solvability is always possible if \$P\$ is self-adjoint or has constant
coefficients, but non-self-adjointness can destroy that property: Lewy's
example was a simple vector-field with complex, non-constant coefficients
arising in the study of several complex variables.

Almost immediately after that discovery, Hormander provided an explanation
of Lewy's example showing that almost all non-self-adjoint operators are
not locally solvable. That was done by considering the essentially dual
problem of existence of non-propagating singularities. I will give
an elementary quantum mechanical interpretation of these issues of local
solvability and non-propagation of singularities in terms of creation and
annihilation operators.

Finally, I will explain how the non-propagating singularities are the
source of (at least some of) the computational problems in finding
eigenvalues of non-self adjoint matrices arising in discretisation of high
energy or semi-classical operators.
```