Vandermonde matrix, also known as a Vandermonde determinant, is an important concept in linear algebra and mathematics with various applications. It is named after Alexandre-Théophile Vandermonde and has numerous uses in different fields.

One of the key properties related to the Vandermonde matrix is Vandermonde's identity (or Vandermonde's convolution). This identity states that any combination of k objects chosen from a group of (m+n) objects must contain r objects from the first m objects and k-r objects from the remaining n objects, where 0 ≤ r ≤ k. This identity holds for all binomial coefficients, not just the non-negative integers assumed in combinatorial proofs.

The Vandermonde determinant has applications in the representation theory of the symmetric group. When the values belong to a finite field, the Vandermonde determinant is sometimes referred to as a Moore determinant. This determinant has specific properties that are utilized in various areas such as the theory of BCH codes and Reed–Solomon error correction codes.

In addition, the Vandermonde polynomial plays a crucial role in constructing interpolation polynomials. The matrix formed by the coefficients of the Vandermonde polynomial is known as a Vandermonde matrix. However, it is important to note that the condition number of the Vandermonde matrix can be significant, potentially leading to large errors when computing the coefficients using methods like Gaussian elimination.

Comparing Vandermonde formula and Lagrange formula, Lagrange formula is typically preferred when the goal is to compute the value of the polynomial p(x) at a given x rather than determining the polynomial coefficients. In such cases, Lagrange formula offers a complexity of O(n^2), making it more efficient for certain computational tasks than the Vandermonde formula.