# THEMATIC PROGRAMS

April 24, 2017

## Numerical and Computational Challenges in Science and Engineering Program

### Short Course on Matrix Valued Function Theory

#### Instructor: Olavi Nevanlinna, Helsinki University of Technology

 Housing and Hotels Thematic Year Homepage Visitor Information

Schedule:
Dates: October 11, 18, 19, 25 and 26th, 2001
Times: 10:00 am-12:00 pm
Location: The Fields Institute, Room 210

Short description:
The course intends to present a new tool for perturbation analysis of matrices. If a large matrix is perturbed with a low rank matrix the spectrum can change dramatically and therefore the resolvent, the eigenvalues being its poles, changes
dramatically if considered as an analytic function. In contrast, if considered as a meromorphic function the perturbation
is small.

The "tool" T1 is presented in:

Olavi Nevanlinna, Growth of operator valued meromorphic functions, Annales Academiae Sci.
Fenn. Math., Vol 25, 2000, 3-30.

The original article discusses operators in Hilbert spaces but the lectures shall concentrate on matrices. It shall be demonstrated how the tool can be used e.g. in deriving error bounds for Krylov solvers which are robust in low rank perturbations.

Preliminary content of lectures:
1. Resolvent of a matrix as a meromorphic function. An example of a compact operator
with spectrum at origin which transforms to a self adjoint operator with a rank-
one perturbation.
2. (2-3) Two lectures on basics of value distribution theory for scalar meromorphic
functions. In particular, main properties of the characteristic function T.
3. (2-3) Two lectures on basics of value distribution theory for scalar meromorphic
functions. In particular, main properties of the characteristic function T.
4. $T_\infty$ and T1: Two generalizations of T to matrix and operator valued
functions.
5. The total logarithmic size $s(A):=\sum \log^+ \sigma_j$ (where $\sigma_j$ denote
the singular values) of a matrix. Basic properties like behavior when forming sum,
product, Kronecker product etc. Behavior in similarity transformations.
6. Subharmonicity of the total logarithmic size s(A(z)) of a matrix valued analytic
function A(z). Behavior near possible poles.
7. Perturbation results for the resolvent when "small" means small rank (but not small
norm). Defective eigenvalues (in the sense of linear algebra) are linked to Picard
exceptional values and to defects in value distribution theory.
8. Application to Krylov subspace solvers: convergence bounds which are robust in low
rank changes of the iteration matrix.
9. Applications to power bounded operators, e.g. to the behavior of An+1-An to
Kreiss matrix theorem etc.
10. What next?
For more details on the thematic year see the link on the Program Page