# Definition of quadratic form

Put content here.## Definition of Quadratic Form
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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.
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### Definition
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Given a symmetric n x n matrix A and vector x in R^n, the quadratic form associated with A is:
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**q_A(x) = x^T A x = sum over i,j of a_ij * x_i * x_j**
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Equivalently, q_A(x) = x dot (Ax).
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### Examples
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For n = 2, with A = [[1, 2], [2, 3]]:
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q(x1, x2) = x1^2 + 4*x1*x2 + 3*x2^2
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For n = 3, with A = I (identity):
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q(x1, x2, x3) = x1^2 + x2^2 + x3^2
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### Classification
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A quadratic form is classified by the eigenvalues of its matrix A:
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- **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
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### Principal Axes Theorem
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By the Spectral Theorem, every symmetric matrix A can be diagonalized by an orthogonal matrix Q:
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**Q^T A Q = D** (diagonal)
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In the new coordinates y = Q^T x, the quadratic form becomes:
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**q(y) = lambda_1 * y1^2 + lambda_2 * y2^2 + ... + lambda_n * yn^2**
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### Applications
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- **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

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