What is a Quadratic Form?
A quadratic form is an expression involving a quadratic polynomial of several variables. It is often represented using matrices for simplification and analysis.
For example, the expression:
Q(x) = 3x² + 4xy + 5y²
can be written in matrix form as:
Q(x) = [x y] × [[3, 2], [2, 5]] × [x y]ᵗ
💡 Notice how the cross term (4xy) becomes 2 in both symmetric off-diagonal positions: 2 + 2 = 4.
General Matrix Representation:
For a vector x = [x₁, x₂, ..., xₙ] and symmetric matrix A,
the quadratic form is:
Q(x) = xᵗAx
Classification of Quadratic Forms
Based on the nature of Q(x) = xᵗAx, quadratic forms are classified as:
-
Positive Definite: Q(x) > 0 for all non-zero x.
💡 All eigenvalues of A are positive. -
Negative Definite: Q(x) < 0 for all non-zero x.
💡 All eigenvalues of A are negative. -
Positive Semi-Definite: Q(x) ≥ 0 for all x.
💡 All eigenvalues are ≥ 0. -
Negative Semi-Definite: Q(x) ≤ 0 for all x.
💡 All eigenvalues are ≤ 0. -
Indefinite: Q(x) takes both positive and negative values.
💡 Eigenvalues have mixed signs.
Applications of Quadratic Forms
- Optimization: In quadratic programming, the goal is to minimize or maximize a quadratic form subject to constraints.
- Machine Learning: In algorithms like Support Vector Machines, kernels and margins use quadratic expressions.
- Physics: In energy expressions like potential energy stored in springs.
- Statistics: Used in analyzing variance and in least squares estimation.