Vandermonde Determinant

Look, if we must discuss this, let's get it over with. In the grand, often tedious, theater of linear algebra, certain constructs appear with an air of self-importance. One such case is the determinant of a Vandermonde matrix, a value referred to—with a stunning lack of imagination—as the Vandermonde determinant.

Named after Alexandre-Théophile Vandermonde, who likely had better things to do than ponder matrices that weren't even formally invented yet, this determinant is a surprisingly elegant expression. It appears in various fields, usually showing up uninvited to solve problems in polynomial interpolation, coding theory, and other areas where you desperately need a unique solution and can't be bothered to find it yourself. Its primary, and frankly, most interesting feature is its direct relationship to the distinctness of the elements that define it. If you're looking for a mathematical litmus test for whether your numbers are unique, you could do worse. You could also do better, but this is what we're discussing today.

Definition

Let's not draw this out. A Vandermonde matrix is a specific type of matrix that arises from a geometric progression in each row. For a set of numbers, or more formally, elements { $α_1, α_2, \dots, α_n$ } from a commutative ring, the square Vandermonde matrix is defined as:

$V = \begin{bmatrix} 1 & \alpha_1 & \alpha_1^2 & \dots & \alpha_1^{n-1} \\ 1 & \alpha_2 & \alpha_2^2 & \dots & \alpha_2^{n-1} \\ 1 & \alpha_3 & \alpha_3^2 & \dots & \alpha_3^{n-1} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 1 & \alpha_n & \alpha_n^2 & \dots & \alpha_n^{n-1} \end{bmatrix}$

The Vandermonde determinant is simply the determinant of this matrix, denoted as det(V). It has a remarkably tidy closed-form expression, which is arguably its only redeeming quality:

$\det(V) = \prod_{1 \le i < j \le n} (\alpha_j - \alpha_i)$

This is the product of all possible differences (α_j - α_i) where j is strictly greater than i. For a small, manageable case like n=3, this expands to (α_2 - α_1)(α_3 - α_1)(α_3 - α_2). Simple. Almost insultingly so. This formula saves you the soul-crushing labor of calculating the determinant using a cofactor expansion, a favor for which you should be quietly grateful.

Main Properties

If you were paying attention to the formula—a significant 'if'—the core property of the Vandermonde determinant should be screamingly obvious.

Non-vanishing Condition: The determinant det(V) is non-zero if and only if all the elements { $α_1, α_2, \dots, α_n$ } are distinct. If any two elements are the same, say α_i = α_j for i ≠ j, then one of the terms in the product (α_j - α_i) becomes zero. And since anything multiplied by zero is zero—a concept I trust you've grasped—the entire determinant vanishes. Geometrically, this means the corresponding rows of the matrix are identical, rendering the matrix singular and its determinant precisely zero. This makes the Vandermonde determinant a convenient, if dramatic, tool for testing uniqueness.
Polynomial Interpolation: This is where the determinant pretends to be useful. Consider the problem of finding a polynomial P(x) of degree at most n-1 that passes through n distinct points (α_1, y_1), (α_2, y_2), ..., (α_n, y_n). This sets up a system of linear equations for the coefficients of the polynomial. The matrix of this system is precisely the Vandermonde matrix V. The existence of a unique solution to this interpolation problem is guaranteed if and only if the matrix is invertible, which, as we've just established, happens if and only if the α_i are all distinct. The non-zero determinant is the universe's way of telling you that, yes, you can draw exactly one parabola through three points. Groundbreaking. The solution itself is often expressed as a Lagrange polynomial.
Relation to the Discriminant: For a monic polynomial p(x) = (x - α_1)(x - α_2)...(x - α_n), the discriminant Δ is defined as Δ = ∏(α_j - α_i)² for 1 ≤ i < j ≤ n. A quick glance reveals that the discriminant is simply the square of the Vandermonde determinant: Δ = (det(V))². This links the geometry of matrix transformations to the algebraic properties of polynomial roots. A cute, tidy relationship.

Proof of the Formula

There are several ways to prove the determinant formula. Most involve a tedious application of row operations, which is brutish and lacks finesse. A more elegant proof, if you're into that sort of thing, treats the determinant as a polynomial. Try to keep up.

Let V_n(α_1, ..., α_n) be the Vandermonde determinant. Consider it as a polynomial in the variable α_n, with the other α_i held constant. The entries in the last row are 1, α_n, α_n², ..., α_n^(n-1). By using cofactor expansion along this last row, we can see that V_n is a polynomial in α_n of degree at most n-1.
Now, suppose we set α_n = α_i for some i < n. In this case, the n-th row of the matrix becomes identical to the i-th row. A fundamental property of determinants is that if two rows are identical, the determinant is zero. Therefore, V_n is zero when α_n is equal to any of α_1, α_2, ..., α_(n-1).
This means that α_1, α_2, ..., α_(n-1) are the roots of the polynomial V_n(α_n). According to the factor theorem, if r is a root of a polynomial P(x), then (x-r) is a factor of P(x). Consequently, V_n must be divisible by (α_n - α_1), (α_n - α_2), ..., (α_n - α_(n-1)).
We can write V_n = C * ∏(α_n - α_i) for i=1 to n-1, where C is some factor that depends on α_1, ..., α_(n-1). The term C is precisely the cofactor of the α_n^(n-1) term in the original matrix, which is the Vandermonde determinant of the matrix formed by the first n-1 rows and columns. That is, C = V_(n-1)(α_1, ..., α_(n-1)).
By applying this argument recursively (an induction, for those of you who enjoy formal suffering), we arrive at the formula: $\det(V_n) = \prod_{1 \le i < j \le n} (\alpha_j - \alpha_i)$

The leading coefficient is 1, which can be confirmed by inspecting the coefficient of the monomial α_2 * α_3² * ... * α_n^(n-1), which comes from the main diagonal of the matrix. It's 1. The proof is complete. You may now resume your regularly scheduled ennui.

Applications

Despite its purely academic feel, this determinant does occasionally escape the chalkboard and find work.

Error-Correcting Codes: In coding theory, specifically in the construction of Reed–Solomon codes, Vandermonde matrices are used to define the generator matrix. The non-vanishing property of the determinant over a finite field ensures that a certain number of errors can be detected and corrected. It's a cog in a much larger, more interesting machine.
Quantum Mechanics: The Slater determinant in quantum mechanics, which describes the wavefunction for a system of multiple identical fermions, often takes the form of a Vandermonde determinant (or a related structure called an alternant). This ensures the wavefunction is antisymmetric with respect to the exchange of two particles, a requirement known as the Pauli exclusion principle. Apparently, even fundamental particles have to obey the rules of linear algebra. A tragic state of affairs.
Statistics: In statistics, the Vandermonde determinant appears in the regression analysis of polynomial models, where it relates to the independence of the regressors. Its non-zero value is critical for the model to be well-defined.