Ke Ye University of Chicago The matrix decomposition problem (MDP) concerns the possibility and feasibility of decomposing an arbitrary (generic) matrix into the product of matrices of a given type. For instance, LU, QR and SVD decompositions are well known examples of the MDP. The most important goal of the MDP is to find a better algorithm to solve a linear system. I will introduce the method of applying algebraic geometry to solve the MDP. I will first derive some general results using algebraic geometry and then apply them to some specific examples.