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
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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
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
When two or more αi 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 αi = αi + 1 = ... = αi+k and αi ≠ αi − 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
Properties
In the case of a square Vandermonde matrix, the Leibniz formula for the determinant gives
where Sn denotes the set of permutations of
Each of these factors must divide the determinant, because the latter is an alternating polynomial in the n variables. It also follows that the Vandermonde determinant divides any other alternating polynomial; the quotient will be a symmetric polynomial.
If m ≤ n, then the matrix V has maximum rank (m) if and only if all αi are distinct. A square Vandermonde matrix is thus invertible if and only if the αi 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
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 uj of the polynomial(s)
of degree ≤ n − 1 which has (have) the property
The Vandermonde matrix can 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
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 αi are chosen to be the m different mth roots of unity.
The Vandermonde matrix diagonalizes the companion matrix.
The Vandermonde matrix is used in some forms of Reed–Solomon error correction codes.