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*m*×*n*matrix
or

for all indices

*i*and*j*. (Some authors use the transpose of the above matrix.)
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 α_{i}are distinct, then it is non-zero.
The Vandermonde determinant is sometimes called the discriminant, although many sources, including this article, refer to the discriminant as the square of this determinant. Note that the Vandermonde determinant is

*alternating*in the entries, meaning that permuting the α_{i}by an odd permutation changes the sign, while permuting them by an even permutation does not change the value of the determinant. It thus depends on the order, while its square (the discriminant) does not depend on the order.
When two or more α

_{i}are equal, the corresponding polynomial interpolation problem (see below) is underdetermined. In that case one may use a generalization called**confluent Vandermonde matrices**, which makes the matrix non-singular while retaining most properties. If α_{i}= α_{i + 1}= ... = α_{i+k}and α_{i}≠ α_{i − 1}, then the (*i*+*k*)th row is given by
The above formula for confluent Vandermonde matrices can be readily derived by letting two parameters α

_{i}and α_{j}go arbitrarily close to each other. The difference vector between the rows corresponding to α_{i}and α_{j}scaled to a constant yields the above equation (for*k*= 1). Similarly, the cases*k*> 1 are obtained by higher order differences. Consequently, the confluent rows are derivatives of the original Vandermonde row.**Properties**

Using the Leibniz formula for the determinant,

where

*S*_{n}denotes the set of permutations of , and sgn(σ) denotes the signature of the permutation*σ*, we can rewrite the Vandermonde determinant as
The Vandermonde polynomial (multiplied with the symmetric polynomials) generates all the alternating polynomials.

If

*m*≤*n*, then the matrix*V*has maximum rank (*m*) if and only if all α_{i}are distinct. A square Vandermonde matrix is thus invertible if and only if the α_{i}are distinct; an explicit formula for the inverse is known.**Applications**

The Vandermonde matrix

*evaluates*a polynomial at a set of points; formally, it transforms*coefficients*of a polynomial to the*values*the polynomial takes at the points α_{i}. The non-vanishing of the Vandermonde determinant for distinct points α_{i}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 unique solution; this result is called the*unisolvence theorem.*
They are thus useful in polynomial interpolation, since solving the system of linear equations

*Vu*=*y*for*u*with*V*an*m*×*n*Vandermonde matrix is equivalent to finding the coefficients*u*_{j}of the polynomial(s)
of degree ≤

*n*− 1 which has (have) the property
The Vandermonde matrix can easily be inverted in terms of Lagrange basis polynomials:each

*column*is the coefficients of the Lagrange basis polynomial, with terms in increasing order going down. The resulting solution to the interpolation problem is called the Lagrange polynomial.
The Vandermonde determinant plays a central role in the Frobenius formula, which gives the character of conjugacy classes of representations of the symmetric group.

When the values α

_{k}range over powers of a finite field, then the determinant has a number of interesting properties: for example, in proving the properties of a BCH code.
Confluent Vandermonde matrices are used in Hermite interpolation.

A commonly known special Vandermonde matrix is the discrete Fourier transform matrix (DFT matrix), where the numbers α

_{i}are chosen equal to the*m*different*m*th roots of unity.
The Vandermonde matrix diagonalizes a companion matrix.

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