Resumen de A Laplacian decomposition algorithm for square matrices
José Alberto Conejero Casares, Antonio Falcó Montesinos, María Mora Jiménez
Many of today’s problems can be posed as a system of equations with the form Ax = b. When A has the form A =d i=1idn1 ⊗ . . . idni−1 ⊗ Ai ⊗ idni+1 ⊗ · · · ⊗ idnd , we say that A is a Laplacian matrix. For example, these matrices appear when discretizing some PDEs, such as the Poisson equation, and are an interesting object of study since the Proper Generalized Decomposition algorithm converges very quickly with the solution of the associated linear system. This made us wonder if a generic square matrix M ∈ GL(RN ) could be decomposed in some way so that the study of the associated linear problem Mx = b would be simpler. In the main theorem of this work, we present a decomposition of the space RN×N that will help us to determine the matrix decomposition we are looking for. In addition, we will show the procedure to be carried out for it in the form of an algorithm.
