The Vandermonde Determinant, A Novel Proof
Applications of the Vandermonde Matrix and An Original Proof of the Determinant
Almost all students of linear algebra will learn about the Vandermonde matrix at some point throughout the course of their studies. This type of matrix has wide applications in math and science and it is quite accessible, which makes it a very useful tool in your mathematical tool-belt. In this article, we'll take a look at a simple application of the Vandermonde matrix to interpolation, and then conclude with an original proof of the well-known determinant of the Vandermonde matrix. First let’s start with a simple definition.
Definition: A Vandermonde matrix is a square matrix of the form
Perhaps the most common application of the Vandermonde matrix is in the area of interpolation. Suppose we have a collection of n points in the plane
We can connect these points with a smooth curve, in particular, with a polynomial of degree n-1:
To do this, we plug our points into the polynomial which produces the system
But note that this is equivalent to the matrix equation
We should be delighted to see our Vandermonde matrix there on the left. So, we have reduced this to solving a matrix equation as in the above. Alas, not all matrix equations admit solutions. How will we know the above equation is solvable? One way is to verify that the Vandermonde matrix will have a non-zero determinant.
It happens that the Vandermonde determinant is something of a celebrity in Linear Algebra. The expression for the determinant is surprisingly elegant, as we’ll see in just a moment, and it seems like everyone has their own way of proving it. This is exactly what we are going to do here, we are going to write an original proof — as near as I can tell — of the determinant of the Vandermonde matrix.
Theorem If A is a Vandermonde matrix then
Proof (by induction) We proceed by induction on the order, n, of the matrix. If n=1 there is nothing to show. In the spirit of verification, let n=2. Then
Now, in this special case the formula for the determinant is quite simple to use, and produces
as desired. Now suppose the claim holds for n-1. Let A be a Vandermonde matrix of order n. The, by row operations we get
and, by Cauchy’s Theorem, we get that
again, by Cauchy, we have that
and by the inductive hypothesis, we get
as desired.
Now, equipped with an expression of the determinant, it is easy to see if the original matrix equation, above, has a solution. In fact, what we can show is that since all the x-values of our points are distinct, indeed, there will be a solution. This is just one of the many ways the Vandermonde matrix is made useful.
