Math Visualization List
Calculus
First Derivative Second Derivative Integration
Determinant
2x2 & 3x3 Determinants Cramer's Rule Permutations & Inversions Permutation Definition & Expansion Properties of Determinants: Row Operations Minors & Cofactors Two-Row Laplace Expansion Vandermonde Determinant
Matrix Operations
Matrix Addition, Subtraction & Scalar Multiplication Matrix Multiplication & Power Transpose, Determinant & Adjoint Inverse Matrix & Matrix Equations Elementary Transformations, Echelon & Canonical Form
Probability & Statistics
Conditional Probability, Independence & Total Probability Discrete Probability Distributions PMF & CDF 2D Random Vectors Conditional Probability Distributions Bayes' Theorem Numerical Characteristics of Random Variables Higher Moments & Covariance Matrix Law of Iterated Expectations Independence, Mean Independence & Uncorrelated Continuous Statistical Distributions
Vector Operations
Vector Addition, Subtraction & Scalar Multiplication Linear Combination, Span & Basis Matrices as Transformations Composition of Transformations 3D Linear Transformations Determinant Inverse, Column Space & Rank Cross-Dimensional Transformations Vector Angle Cross Product Change of Basis Eigenvectors & Eigenvalues Function Vector Spaces Geometric Interpretation of Cramer's Rule Gravity Simulation

Vandermonde Determinant: Compression Formula Demo

Difference Product Expansion Steps

Step Factor Current Product

Basic Introduction

A Vandermonde matrix is defined as:

V = [x_i^(j-1)] (i,j=1..n)

Its determinant has a classic compression formula:

det(V) = ∏(x_j - x_i), 1≤i<j≤n

Interpretation

  • If x_i = x_j for some i≠j, then one factor is 0, and the determinant is 0.
  • If all x_i are distinct, the determinant is non-zero.
  • This structure is commonly used in interpolation, numerical analysis, and polynomial basis transformation.