Module IV·Article III·~2 min read
Bilinear and Quadratic Forms
Linear Operators and the Jordan Form
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Quadratic Forms
A quadratic form is a homogeneous polynomial of degree 2: $Q(x) = \sum_{ij} a_{ij} x_i x_j = x^T A x$, where $A$ is a symmetric matrix ($A = A^T$).
Examples: $Q(x, y) = x^2 + 2 x y + 3 y^2$ corresponds to $A = \begin{bmatrix} 1 & 1 \ 1 & 3 \end{bmatrix}$.
Sylvester's Law of Inertia: Any quadratic form over $\mathbb{R}$ can be reduced by a change of variables to the form $x_1^2 + \ldots + x_p^2 - x_{p+1}^2 - \ldots - x_{p+i}^2$ ($p$ positive, $i$ negative, the rest zero). The numbers $p$ and $i$ do not depend on the choice of substitution.
Classification of Real Forms
Positive definite: $Q(x) > 0$ for all $x \ne 0$ $\iff$ all eigenvalues of $A$ are
gt; 0$ $\iff$ all leading principal minors gt; 0$ (Sylvester's criterion).Negative definite: $Q(x) < 0$ $\iff$ all eigenvalues
lt; 0$.Indefinite: There are both positive and negative eigenvalues.
Semidefinite: Some eigenvalues are zero.
Reduction to Canonical Form
Method of completing the square: successively eliminate cross terms.
$x^2 + 4 x y + 5 y^2 = (x + 2y)^2 + y^2$.
Matrix method: an orthogonal matrix $P$ transforms $A$ to diagonal form $D = P^T A P = \operatorname{diag}(\lambda_1,...,\lambda_n)$. In the new basis $Q = \lambda_1 y_1^2 + \ldots + \lambda_n y_n^2$.
Applications
Optimization: The second order condition for a minimum is that the Hessian is positive definite.
Geometry: The form $D x^2 + 2 E x y + F y^2 = 1$ defines an ellipse (if the form is positive definite), a hyperbola (indefinite), or a parabola (semidefinite).
Physics: Kinetic energy $T = \frac{1}{2} \dot{x}^T M \dot{x}$ ($M$ is the mass matrix), potential $V = \frac{1}{2} x^T K x$ ($K$ is the stiffness matrix) — quadratic forms. Normal modes are eigenvectors of the system.
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