Related Eigenproblems.

Singular pencils arise naturally in the study of linear control systems $E \dot{x} (t) = Ax(t) + Bu(t)$, $y(t) = Cx(t)$. Here $u(t)$ is a control input that the user attempts to select to make sure the state variable $x(t)$ and output variable $y(t)$ attain desired values. Concepts such as controllability, controllable subspace, controllable and uncontrollable modes, observability, etc. can all be formulated and computed in terms of reducing subspaces and eigenvalues [460,447,451,450].