Classifying Patterns for Commutative Matrix Completion

Matrix Completion Problems explore whether partial matrices having some unspecified entries can be completed in a strategic way so that the partial matrix has a certain property. We focused on the question: under what circumstances can the unspecified entries of a partial matrix X be chosen so that X commutes with a fully specified matrix A? Or, when can X be completed so that AX=XA? Using a Polynomial Approach and a Matrix Equation Approach used in earlier research as well as a less developed Graph Theory Approach, our research group created and proved a Classification Theorem. The theorem gives all the admissible patterns of specified entries for a partial matrix X that allow it to be completed to commute with a matrix A when A is a Jordan Block. We generalize this result to obtain admissible patterns for matrices in Jordan Canonical form, or matrices with multiple Jordan Blocks. Using the resulting admissible patterns from the Classification Theorem, new admissible patterns for partial matrices that commute with a matrix A permutation similar to a Jordan Block are easily determined. This research on commutative matrix completion was conducted over the summer of 2011.