**MSC:**- 65F15 Eigenvalues, eigenvectors
- 93B05 Controllability

We study the inverse power method well-known in numerical linear

algebra from a control point of view. In particular, controllability

properties of the inverse power method on projective space are

investigated. It is known that for complex eigenvalue shifts a simple

characterization of the reachable sets in terms of invariant subspaces

can be obtained. In contrast, the real case under consideration in

this talk is more complicated. Using properties of universally

regular controls, necessary and sufficient conditions for complete

controllability are obtained in terms of the solvability of a matrix

equation. Partial results on conditions for the solvability of this

matrix equation are given.