Partial Differential Equations/The Malgrange-Ehrenpreis theorem

Vandermonde's matrixEdit

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:

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  .


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 theoremEdit

Lemma 10.3:

Let   be pairwise different. The solution to the equation


is given by

 ,  .


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.