This is non-zero if and only if all are distinct (no two are equal), making the Vandermonde matrix invertible.
The polynomial interpolation problem is to find a polynomial which satisfies for given data points . This problem can be reformulated in terms of linear algebra by means of the Vandermonde matrix, as follows. computes the values of at the points via a matrix multiplication , where is the vector of coefficients and is the vector of values (both written as column vectors):
If and are distinct, then V is a square matrix with non-zero determinant, i.e. an invertible matrix. Thus, given V and y, one can find the required by solving for its coefficients in the equation :
which is non-zero if and only if all are distinct.
The Vandermonde determinant was formerly sometimes called the discriminant, but in current terminology the discriminant of a polynomial is the square of the Vandermonde determinant of the roots. The Vandermonde determinant is an alternating form in the , meaning that exchanging two changes the sign, and thus depends on order for the . By contrast, the discriminant does not depend on any order, so that Galois theory implies that the discriminant is a polynomial function of the coefficients of .
By the Leibniz formula, is a polynomial in the , with integer coefficients. All entries of the th column (zero-based) have total degree. Thus, again by the Leibniz formula, all terms of the determinant have total degree
where is a polynomial. As the product of all and have the same degree , the polynomial is, in fact, a constant. This constant is one, because the product of the diagonal entries of is , which is also the monomial that is obtained by taking the first term of all factors in This proves that
Second proof: linear maps
Let F be a field containing all and the Fvector space of the polynomials of degree less than or equal to n with coefficients in F. Let
The Vandermonde matrix is the matrix of with respect to the canonical bases of and
Changing the basis of amounts to multiplying the Vandermonde matrix by a change-of-basis matrix M (from the right). This does not change the determinant, if the determinant of M is 1.
The polynomials , , , …, are monic of respective degrees 0, 1, …, n. Their matrix on the monomial basis is an upper-triangular matrixU (if the monomials are ordered in increasing degrees), with all diagonal entries equal to one. This matrix is thus a change-of-basis matrix of determinant one. The matrix of on this new basis is
Thus Vandermonde determinant equals the determinant of this matrix, which is the product of its diagonal entries.
This proves the desired equality. Moreover, one gets the LU decomposition of V as .
Third proof: row and column operations
This third proof is based on the fact that if one adds to a column of a matrix the product by a scalar of another column then the determinant remains unchanged.
So, by subtracting to each column – except the first one – the preceding column multiplied by , the determinant is not changed. (These subtractions must be done by starting from last columns, for subtracting a column that has not yet been changed). This gives the matrix
As all the entries in the -th row of have a factor of , one can take these factors out and obtain
where is a Vandermonde matrix in . Iterating this process on this smaller Vandermonde matrix, one eventually gets the desired expression of as the product of all such that .
Rank of the Vandermonde matrix
An m × n rectangular Vandermonde matrix such that m ≤ n has rankmif and only if all xi are distinct.
An m × n rectangular Vandermonde matrix such that m ≥ n has rankn if and only if there are n of the xi that are distinct.
A square Vandermonde matrix is invertible if and only if the xi are distinct. An explicit formula for the inverse is known (see below).
Inverse Vandermonde matrix
As explained above in Applications, the polynomial interpolation problem for satisfying is equivalent to the matrix equation , which has the unique solution . There are other known formulas which solve the interpolation problem, which must be equivalent to the unique , so they must give explicit formulas for the inverse matrix . In particular, Lagrange interpolation shows that the columns of the inverse matrix
are the coefficients of the Lagrange polynomials
where . This is easily demonstrated: the polynomials clearly satisfy for while , so we may compute the product , the identity matrix.
Confluent Vandermonde matrices
As described before, a Vandermonde matrix describes the linear algebra interpolation problem of finding the coefficients of a polynomial of degree based on the values , where are distinct points. If are not distinct, then this problem does not have a unique solution (and the corresponding Vandermonde matrix is singular). However, if we specify the values of the derivatives at the repeated points, then the problem can have a unique solution. For example, the problem
where , has a unique solution for all with . In general, suppose that are (not necessarily distinct) numbers, and suppose for simplicity that equal values are adjacent:
where and are distinct. Then the corresponding interpolation problem is
The corresponding matrix for this problem is called a confluent Vandermonde matrix, given as follows. If , then for a unique (denoting ). We let
This generalization of the Vandermonde matrix makes it non-singular, so that there exists a unique solution to the system of equations, and it possesses most of the other properties of the Vandermonde matrix. Its rows are derivatives (of some order) of the original Vandermonde rows.
Another way to derive this formula is by taking a limit of the Vandermonde matrix as the 's approach each other. For example, to get the case of , take subtract the first row from second in the original Vandermonde matrix, and let : this yields the corresponding row in the confluent Vandermonde matrix. This derives the generalized interpolation problem with given values and derivatives as a limit of the original case with distinct points: giving is similar to giving for small . Geometers have studied the problem of tracking confluent points along their tangent lines, known as compacitification of configuration space.