ABSTRACT
Algorithms for computing invariant subspaces are interpreted using a general framework based upon the fact that the invariant subspace problem is equivalent to solving a system of nonlinear decoupling equations. Computing an invariant subspace relative to a cluster of eigenvalues is often desirable because the individual eigenvalue problems can be ill-conditioned if the eigenvalues are close together and/or the eigenvectors are almost linearly dependent. The nonlinearity of the general process is at the core of interpretation of the invariant subspace problem. There is an important distinction to be made between using the Riccati transformation to compute an invariant subspace and the closely related use of Gaussian transformations such as with treppeniteration. A potential drawback of an iterative scheme to solve the Riccati equation is the dependence on having a good initial guess.
