Definition of quadratic form

Definition of Quadratic Form

A quadratic form is a homogeneous polynomial of degree 2 in n variables. It is a fundamental construction in linear algebra with applications in optimization, physics, and geometry.

Definition

Given a symmetric n x n matrix A and vector x in R^n, the quadratic form associated with A is:

q_A(x) = x^T A x = sum over i,j of a_ij * x_i * x_j

Equivalently, q_A(x) = x dot (Ax).

Examples

For n = 2, with A = [[1, 2], [2, 3]]:

q(x1, x2) = x1^2 + 4x1x2 + 3*x2^2

For n = 3, with A = I (identity):

q(x1, x2, x3) = x1^2 + x2^2 + x3^2

Classification

A quadratic form is classified by the eigenvalues of its matrix A:

• Positive definite: all eigenvalues > 0 (q(x) > 0 for all x != 0)
• Negative definite: all eigenvalues < 0
• Positive semidefinite: all eigenvalues >= 0
• Indefinite: both positive and negative eigenvalues exist

Principal Axes Theorem

By the Spectral Theorem, every symmetric matrix A can be diagonalized by an orthogonal matrix Q:

Q^T A Q = D (diagonal)

In the new coordinates y = Q^T x, the quadratic form becomes:

q(y) = lambda_1 * y1^2 + lambda_2 * y2^2 + ... + lambda_n * yn^2

Applications

• Optimization: Hessian matrix quadratic forms determine local minima/maxima
• Physics: kinetic energy T = (1/2) * v^T M v (mass matrix M)
• Statistics: Mahalanobis distance d^2 = (x - mu)^T Sigma^(-1) (x - mu)
• Conic sections: ellipse, hyperbola, parabola classified by quadratic form signature