  The focus of this paper is the connection between two foundational areas of
LTI systems theory: geometric control and eigenstructure assignment. In
particular, we study the properties of the null-spaces of the reachability
matrix pencil and of the Rosenbrock system matrix, which have been extensively
used as two computational building blocks for the calculation of pole placing
state feedback matrices and pole placing friends of output-nulling subspaces.
Our objective is to show that the subspaces in the chains of kernels obtained
in the construction of these feedback matrices interact with each other in ways
that are entirely independent from the choice of eigenvalues. So far, these
chains of subspaces have only been studied in the case of stationarity. In this
case, it is known that these chains converge to the classic Kalman reachable
subspace for the reachability matrix pencil and to the largest reachability
subspace in the case of the Rosenbrock matrix, respectively. Here we are
interested in showing that even before stationarity has been reached, the
partial chains are linked to structural properties of the system, and are
therefore independent of the closed-loop eigenvalues that we wish to assign. We
further characterize these subspaces by investigating the notion of largest
subspace on which it is possible to assign the closed-loop spectrum (possibly
maintaining the output at zero) without resorting to non-trivial Jordan forms.
