Partial Differential Equations/The Malgrange-Ehrenpreis theorem

Vandermonde's matrix 0 100% #FFFFFF Definition 10.1: Let and let . Then the Vandermonde matrix associated to is defined to be the matrix . For pairwise different (i. e. for ) matrix is invertible, as the following theorem proves: 0 100% #FFFFFF Theorem 10.2: Let be the Vandermonde matrix associated to the pairwise different points . Then the matrix whose -th entry is given by is the inverse matrix of . Proof: We prove that , where is the identity matrix. Let . We first note that, by direct multiplication, . Therefore, if is the -th entry of the matrix , then by the definition of matrix multiplication . The Malgrange-Ehrenpreis theorem 0 100% #FFFFFF Lemma 10.3: Let be pairwise different. The solution to the equation is given by , . Proof: We multiply both sides of the equation by on the left, where is as in theorem 10.2, and since is the inverse of , we end up with the equation . Calculating the last expression directly leads to the desired formula. Exercises Sources