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  .

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 theoremEdit

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. 

ExercisesEdit

SourcesEdit