## 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 + 4*x1*x2 + 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