The common invariant subspace problem: an approach via Gröbner bases

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Let A be an n × n matrix. It is a relatively simple process to construct a homogeneous ideal (generated by quadrics) whose associated projective variety parametrizes the one-dimensional invariant subspaces of A . Given a finite collection of n × n matrices, one can similarly construct a homogeneous ideal (again generated by quadrics) whose associated projective variety parametrizes the one-dimensional subspaces which are invariant subspaces for every member of the collection. Gröbner basis techniques then provide a finite, rational algorithm to determine how many points are on this variety. In other words, a finite, rational algorithm is given to determine both the existence and quantity of common one-dimensional invariant subspaces to a set of matrices. This is then extended, for each d , to an algorithm to determine both the existence and quantity of common d -dimensional invariant subspaces to a set of matrices.