Vandermondedetermináns

The Vandermonde determinant is a specific type of determinant of a square matrix known as a Vandermonde matrix. A Vandermonde matrix is defined by a sequence of elements, say $x_1, x_2, \dots, x_n$. The matrix $V$ has entries $V_{ij} = x_i^{j-1}$ for $1 \le i, j \le n$. Therefore, the matrix looks like this:

$$

V = \begin{pmatrix}

1 & x_1 & x_1^2 & \dots & x_1^{n-1} \\

1 & x_2 & x_2^2 & \dots & x_2^{n-1} \\

\vdots & \vdots & \vdots & \ddots & \vdots \\

1 & x_n & x_n^2 & \dots & x_n^{n-1}

\end{pmatrix}

$$

The determinant of this matrix, denoted as $\det(V)$ or $|V|$, is given by the product of all possible differences of pairs of the $x_i$ values:

$$

\det(V) = \prod_{1 \le i < j \le n} (x_j - x_i)

$$

This formula is often referred to as the Vandermonde determinant formula. An important property of the Vandermonde determinant is that it is non-zero if and only if all the elements $x_1, x_2, \dots, x_n$ are distinct. This property makes Vandermonde matrices and their determinants useful in various areas of mathematics and computer science.

The Vandermonde determinant arises in polynomial interpolation, specifically in Lagrange interpolation. It also appears in the study of symmetric polynomials and in coding theory. The formula for the determinant can be proven by induction, often by performing row operations to simplify the matrix or by considering the determinant as a polynomial in one of the variables.