In linear algebra, a Vandermonde matrix, named after Alexandre-Théophile Vandermonde, is a matrix with the terms of a geometric progression matrix

, the jth, for all zero-based. Some authors define the Vandermonde matrix as the transpose of the above matrix.

The determinant of a square) is called a Vandermonde determinant or Vandermonde polynomial. Its value is:

This is non-zero if and only if are distinct (no two are equal), making the Vandermonde matrix invertible.

Applications

The polynomial interpolation. This problem can be reformulated in terms of linear algebra is the vector of values (both written as column vectors):

are distinct, then V is a square matrix with non-zero determinant, i.e. an invertible matrix. Thus, given V and y:. That is, the map from coefficients to values of polynomials is a bijective linear mapping with matrix V, and the interpolation problem has a unique solution. This result is called the unisolvence theorem, and is a special case of the Chinese remainder theorem for polynomials.

In statistics means that the Vandermonde matrix is the design matrix of polynomial regression.

In numerical analysis naïvely by Gaussian elimination results in an algorithm with time complexity O(n3). Exploiting the structure of the Vandermonde matrix, one can use Newton's divided differences method) to solve the equation in O(n2) time, which also gives the UL factorization is ill-conditioned. (See polynomial interpolation.) Using displacement rank operations with the use of fast matrix multiplication algorithms is the matrix multiplication exponent .

The Vandermonde determinant is used in the representation theory of the symmetric group.

belong to a finite field, the Vandermonde determinant is also called the Moore determinant, and has properties which are important in the theory of BCH codes and Reed–Solomon error correction codes.

The discrete Fourier transform is defined by a specific Vandermonde matrix, the DFT matrix are chosen to be th roots of unity. The Fast Fourier transform time. See the article on Multipoint Polynomial evaluation for details.

In the physical theory of the quantum Hall effect, the Vandermonde determinant shows that the Laughlin wavefunction with filling factor 1 is equal to a Slater determinant. This is no longer true for filling factors different from 1 in the fractional quantum Hall effect.

In the geometry of polyhedra-faces of cyclic polytopes

Determinant

The determinant of a square Vandermonde matrix is called a Vandermonde polynomial or Vandermonde determinant. Its value is the polynomial

are distinct.

The Vandermonde determinant was formerly sometimes called the discriminant, but in current terminology the discriminant is the square of the Vandermonde determinant of the roots. The Vandermonde determinant is an alternating form does not depend on any order, so that Galois theory implies that the discriminant is a polynomial function.

The determinant formula is proved below in three ways. The first uses polynomial properties, especially the unique factorization property of multivariate polynomials. Although conceptually simple, it involves non-elementary concepts of abstract algebra. The second proof is based on the linear algebra concepts of change of basis in a vector space and the determinant of a linear map. In the process, it computes the LU decomposition of the Vandermonde matrix. The third proof is more elementary but more complicated, using only elementary row and column operations.

First proof: Polynomial properties

The first proof relies on properties of polynomials.

By the Leibniz formula, with integer-th column have total degree. Thus, again by the Leibniz formula, all terms of the determinant have total degree

(that is, the determinant is a homogeneous polynomial of this degree).

, one gets a matrix with two equal rows, which has thus a zero determinant. Thus, considering the determinant as univariate the factor theorem

that is

, which is also the monomial and finishes the proof.

Second proof: linear maps

Let be a field the vector space of the polynomials of degree at most with coefficients in . Let

be the linear map to the -tuple that is,

1. .

with respect to the canonical bases Changing the basis (from the right). The polynomials are monic of respective degrees 0, 1, …, whose change-of-basis matrix is an upper-triangular matrix on this new basis is thus:

1. .

The determinant of this matrix is the product of its diagonal entries, so the Vandermonde determinant is: This proves the desired equality, as well as giving the LU decomposition.

Third proof: row and column operations

The 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.

, 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

Applying the Laplace expansion formula, with

, one can take these factors out and obtain

1. ,

.

Rank of the Vandermonde matrix

• An rectangular Vandermonde matrix such that has rank if and only if all are distinct.

• An rectangular Vandermonde matrix such that has rank if and only if there are of the that are distinct.

• A square Vandermonde matrix is invertible if and only if the are distinct. An explicit formula for the inverse is known (see below).

Generalizations

is the Vandermonde determinant.

By multiplying with the Hermitian conjugate

Inverse Vandermonde matrix

As explained above in Applications, the polynomial interpolation. In particular, Lagrange interpolation shows that the columns of the inverse matrix

are the coefficients of the Lagrange polynomials , the identity matrix.

Confluent Vandermonde matrices

As described before, a Vandermonde matrix describes the linear algebra interpolation problem are distinct 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

are (not necessarily distinct) numbers, and suppose for simplicity that equal values are adjacent:

are distinct. Then the corresponding interpolation problem is

The corresponding matrix for this problem is called a confluent Vandermonde matrix, given as follows.). 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.

. Geometers have studied the problem of tracking confluent points along their tangent lines, known as compactification of configuration space.

See also

• Schur polynomial – a generalization

• Alternant matrix

• Lagrange polynomial

• Wronskian

• List of matrices

• Moore determinant over a finite field

• Vieta's formulas

References

Further reading

• .