for all indices i and j. The identical term Vandermonde matrix was used for the transpose of the above matrix by Macon and Spitzbart (1958). The Vandermonde matrix used for the Discrete Fourier Transform matrix satisfies both definitions.
The determinant of a square Vandermonde matrix (where m = n) can be expressed as
This is called the Vandermonde determinant or Vandermonde polynomial. If all the numbers are distinct, then it is non-zero.
The Vandermonde determinant was sometimes called the discriminant, although, presently, the discriminant is the square of the Vandermonde determinant. The Vandermonde determinant is an alternating form in the , meaning that exchanging two changes the sign, while permuting the by an even permutation does not change the value of the determinant. It thus depends on the choice of an order on the , while its square, the discriminant, does not depend on any order, and this implies, by Galois theory, that the discriminant is a polynomial function of the coefficients of the polynomial that has the as roots.
The main property of a square Vandermonde matrix
is that its determinant has the simple form
This may be proved either by using properties of polynomials or elementary row and column operations. The former is simpler but it is non-constructive and uses unique factorization property of multivariate polynomials. The latter is constructive and more elementary, at the price of being more complicated. A third proof, based on Gaussian elimination, is sketched. It is still more complicated, if written in details, but provides the U-part of the LU decomposition of Vandermonde matrices.
Using polynomial properties
By Leibniz formula, det(V) is a polynomial in the with integer coefficients. All entries of the ith column have the total degree i – 1. Thus, again by Leibniz formula, all terms of the determinant have the total degree
(that is the determinant is a homogeneous polynomial of this degree).
If, for i ≠ j, one substitutes for , one gets a matrix with two equal columns, which has thus a zero determinant. Thus, by the factor theorem, is a divisor of det(V). By unique factorization property of multivariate polynomials, the product of all divides det(V), that is
where Q is a polynomial. As the product of all and det(V) have the same degree the polynomial Q is, in fact, a constant. This constant is one, because the product of the diagonal entries of V is which is also the monomial that is obtained by taking the first term of all factors in This proves that
By row and column operations
This second proof is based on the fact that, if one adds to a row (or a column) of a matrix the product by a scalar of another row (or column), the determinant remains unchanged.
If one subtracts the first row of V to all the other rows, the determinant is not changed, and the new matrix has the form
where is a row matrix, is a column of zeros, and A is a square matrix, such that
The entry of the (i – 1)th row and the (j – 1)th column of A (that is the ith row and the jth column of the whole matrix) is
Dividing out from the (i – 1)th row of A, for i = 2, ..., n, one gets a matrix B such that
The coefficient of the (i – 1)th row and the (j – 1)th column of B is
for i = 2, ..., n, and setting
Thus, subtracting, for j running from n down to 2, the (j – 2)th column of B multiplied by from the (j – 1)th column, one gets a (n – 1) × (n – 1) Vandermonde matrix in which has the same determinant as B. Iterating this process on this smaller Vandermonde matrix, one gets eventually the desired expression of det(V) as the product of the
By Gaussian elimination, U-part of LU decomposition
The determinant of Vandermonde matrices may also be computed using Gaussian elimination. This provides an explicit form of the upper triangular matrix of the LU decomposition of the matrix. For this computation one uses only the elementary row operations consisting of adding to a row a scalar multiple of a preceding row. This means than one multiplies the matrix by a lower triangular matrix with a diagonal consisting only of 1. In particular, the determinant is not changed by these transformations.
Applying Gaussian elimination to a square Vandermonde matrix, one gets eventually an upper triangular matrix
which has the same determinant as V.
A proof by induction on the steps of Gaussian elimination allows showing that, for 1 ≤ i ≤ j ≤ n, one has
where is an abbreviation for , and is the sum of all monomials of degree d in In particular, the first rows of V and W are equal, and the factor equals 1 for the entries of the diagonal (since 1 is the only monomial of degree 0).
A key ingredient of this proof is that, for k > i
For the recursion, one has to describe the matrix at each step of the Gaussian elimination. Let be the entry of the ith row and jth column of this variable matrix. Before the ith step, the entries that belong either to the i first rows or the i – 1 first columns have the values that they will have at the end of Gaussian elimination, and, for i ≤ j ≤ n and i ≤ h ≤ n, one has
This is true before the first step, and one has to prove that this remains true during Gaussian elimination. The ith step does not change the i first rows nor the i – 1 first columns. It changes to zero for i < h ≤ n. For i < j ≤ n and i < h ≤ n, it changes into That is, the new is
This shows that the structure of the is kept during Gaussian elimination, and thus the form of W.
It follows from the structure of W that is the product of the diagonal entries of W, which proves again the formula for the determinant of a Vandermonde matrix.
A m × n rectangular Vandermonde matrix such that m ≥ n has maximum rank n 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.
The Vandermonde matrix evaluates a polynomial at a set of points; formally, it is the matrix of the linear map that maps the vector of coefficients of a polynomial to the vector of the values of the polynomial at the values appearing in the Vandermonde matrix. The non-vanishing of the Vandermonde determinant for distinct points shows that, for distinct points, the map from coefficients to values at those points is a one-to-one correspondence, and thus that the polynomial interpolation problem is solvable with a unique solution; this result is called the unisolvence theorem.
This may be useful in polynomial interpolation, since inverting the Vandermonde matrix allows expressing the coefficients of the polynomial in terms of the and the values of the polynomial at the . However, the interpolation polynomial is generally easier to compute with the Lagrange interpolation formula, which may be used for deriving a formula for the inverse of a Vandermonde matrix.
The Vandermonde determinant is used in the representation theory of the symmetric group.
When the values belong to a finite field, then the Vandermonde determinant is also called a Moore determinant and has specific properties that are used, for example, in the theory of BCH code and Reed–Solomon error correction codes.
The Laughlin wavefunction with filling factor one (appearing in the Quantum Hall effect), by the formula for the Vandermonde determinant, can be seen to be a Slater determinant. This is not true anymore for filling factors different from one, i.e., in the fractional Quantum Hall effect.
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 (which is reflected by the fact that the corresponding Vandermonde matrix is singular). However, if we give the values of the derivatives at the repeated points, then the problem can have a unique solution. For example, the problem
where is a polynomial of degree , has a unique solution for all . In general, suppose that are (not necessarily distinct) numbers, and suppose for ease of notation that equal values come in continuous sequences in the list. That is
where and are distinct. Then the corresponding interpolation problem is
And the corresponding matrix for this problem is called a confluent Vandermonde matrices. In our case (which is the general case, up to permuting the rows of the matrix) the formula for it is given as follows: if , then for some (unique) (we consider ). Then, we have
This generalization of the Vandermonde matrix makes it non-singular (such that there exists a unique solution to the system of equations) while retaining most properties of the Vandermonde matrix. Its rows are derivatives (of some order) of the original Vandermonde rows.
Another way to receive this formula is to let some of the 's go arbitrarily close to each other. For example, if , then letting in the original Vandermonde matrix, the difference between the first and second rows yields the corresponding row in the confluent Vandermonde matrix. This allows us to link the generalized interpolation problem (given value and derivatives on a point) to the original case where all points are distinct: Being given is similar to being given where is very small.
- Roger A. Horn and Charles R. Johnson (1991), Topics in matrix analysis, Cambridge University Press. See Section 6.1.
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- Macon, N.; A. Spitzbart (February 1958). "Inverses of Vandermonde Matrices". The American Mathematical Monthly. 65 (2): 95–100. doi:10.2307/2308881. JSTOR 2308881.
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- François Viète (1540-1603), Vieta's formulas, https://en.wikipedia.org/wiki/Vieta%27s_formulas
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- Vandermonde matrix at ProofWiki