I’m following the book Mimetic Discretization Methods by Castillo and Miranda, reading the page 209-211, it states the following:
Let be a Vandermonde matrix, of order with generator . Where the order implies the highest power of Vandermonde matrix, therefore has rows and columns.
Let
Generators of the following matrices
Just for sake of example looks like:
The null space of this matrix is given by , that is easily verifiable just by that results in the null vector, as expected.
The other Vandermonde matrices share the same null space. The book then states:
In general, the null space of a -th order Vandermonde matrix has dimension .
My first question: There is no references or proofs to support this claim. Is this true? How can I prove that? What are conditions to that be true. I could not find anywhere this affirmation.
The text continues, and we need to solve the following system
Where are the Vandermonde matrices above, is a unknown vector and
Lets only focus in since the construction to other matrices are equivalent. We then need to solve the following:
completely written as
Then the text declares that
for some .
My second question is: where that vector in first term came from? Is that true? How can I show that?
I’m able to solve that system using least squares, but could not find a correlation.
Some tries: (EDIT)
Definition (Null Space): is a null space of if an only if for any .
Lets say by hypothesys that
Since we are looking for (I will drop the indexes for sake of notation and typing):
by the hypothesis
by Null Space definition
We are just back where we started, seems a tautology in my view.
Anyway I also know that
I can find by least squares, and have by the null space analysis. Therefore I just need to find to uniquely identify .