Vandermonde matrix

In linear algebra, a Vandermonde matrix, named after Alexandre-Théophile Vandermonde, is a matrix with the terms of a geometric progression in each row, i.e., an matrix
or
for all indices and . 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 can be expressed as
This is called the Vandermonde determinant or Vandermonde polynomial. It is non-zero if and only if all formula_4 are distinct.
The Vandermonde determinant was sometimes called the "discriminant", although, presently, the discriminant of a polynomial is the square of the Vandermonde determinant of the roots of the polynomial. The Vandermonde determinant is an alternating form in the formula_4, meaning that exchanging two formula_4 changes the sign, while permuting the formula_4 by an even permutation does not change the value of the determinant. It thus depends on the choice of an order for the formula_4, 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 formula_4 as roots.
Proofs.
The main property of a square Vandermonde matrix
is that its determinant has the simple form
Three proofs of this equality are given below. The first one 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 does not require any explicit computation, but involves the concepts of the "determinant of a linear map" and change of basis. It provides also the structure of the LU decomposition of the Vandermonde matrix. The third one is more elementary and more complicated, using only elementary row and column operations.
Using polynomial properties.
By the Leibniz formula, is a polynomial in the formula_12 with integer coefficients. All entries of the 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).
If, for , one substitutes formula_4 for formula_15, one gets a matrix with two equal rows, which has thus a zero determinant. Thus, by the factor theorem, formula_16 is a divisor of . By the unique factorization property of multivariate polynomials, the product of all formula_16 divides , that is 
where is a polynomial. As the product of all formula_16 and have the same degree formula_20 the polynomial is, in fact, a constant. This constant is one, because the product of the diagonal entries of is 
formula_21
which is also the monomial that is obtained by taking the first term of all factors in formula_22 This proves that 
Using linear maps.
Let be a field containing all formula_12 and formula_25 the vector space of the polynomials of degree less than with coefficients in . Let 
be the linear map defined by 
The Vandermonde matrix is the matrix of formula_28 with respect to the canonical bases of formula_25 and formula_30
Changing the basis of formula_25 amounts to multiplying the Vandermonde matrix by a change-of-basis matrix (from the right). This does not change the determinant, if the determinant of is .
The polynomials formula_32 are monic of respective degrees . Their matrix on the monomial basis is an upper-triangular matrix (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 formula_28 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 as
By 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 formula_36 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 along the first row, we obtain formula_38, with
As all the entries in the formula_40-th row of formula_41 have a factor of formula_42 one can take these factors out and obtain
where formula_44 is a Vandermonde matrix in formula_45 Iterating this process on this smaller Vandermonde matrix, one eventually gets the desired expression of as the product of all formula_16 such that .
Resulting properties.
An rectangular Vandermonde matrix such that has maximum rank if and only if all are distinct. 
An rectangular Vandermonde matrix such that has maximum 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.
Applications.
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 formula_47 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, and is a special case of the Chinese remainder theorem for polynomials.
This may be useful in polynomial interpolation, since inverting the Vandermonde matrix allows expressing the coefficients of the polynomial in terms of the formula_47 and the values of the polynomial at the formula_47.
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 formula_50 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 discrete Fourier transform is defined by a specific Vandermonde matrix, the DFT matrix, where the numbers α"i" are chosen to be roots of unity.
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.
It is the design matrix of polynomial regression.
Confluent Vandermonde matrices.
As described before, a Vandermonde matrix describes the linear algebra interpolation problem of finding the coefficients of a polynomial formula_51 of degree formula_52 based on the values formula_53, where formula_54 are "distinct" points. If formula_47 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 formula_57 is a polynomial of degree formula_58, has a unique solution for all formula_59. In general, suppose that formula_60 are (not necessarily distinct) numbers, and suppose for ease of notation that equal values come in continuous sequences in the list. That is
where formula_62 formula_63 and formula_64 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 formula_66, then formula_67 for some (unique) formula_68 (we consider formula_69). 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 formula_47's go arbitrarily close to each other. For example, if formula_72, then letting formula_73 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 formula_74 is similar to being given formula_75 where formula_76 is very small.