What are the real life applications of quadratic forms? I have used them to sketch conics but are there any other applications?
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 Quadratic forms show up in many places. In physics, energy is often a quadratic form. For example, the kinetic energy of a rigid body is $$T = \frac{1}{2}\mathbf{\omega^T I \omega}$$ where $\mathbf{\omega}$ is the angular velocity vector (3x1) and $\mathbf{I}$ is the tensor of inertia (just think of it as a 3x3 matrix). Often times we want to find the principle axes, which simply means finding a rotation that makes $\mathbf{I}$ diagonal (this is where your eigenvectors matter!), so that the quadratic form becomes a simple sum of squares. Another place they show up is in optimization. Consider a twice-differentiable function of N variables $f(\mathbf{x})$, where $\mathbf{x}$ is the Nx1 vector of variables. If we want to find a local maximum and minimum, these will occur at a location (call it $\mathbf{x=x_0}$) where the first derivatives are zero, $$\left. \nabla f(x) \right|_{\mathbf{x=x_0}} = \mathbf{0}.$$ This equation represents N scalar equations. The first two terms of the Taylor expansion of f about $\mathbf{x=x_0}$ is then $$f(\mathbf{x}) \approx f(\mathbf{x_0}) + \frac{1}{2}(\mathbf{x-x_0})^T \mathbf{H(\mathbf{x_0})}(\mathbf{x-x_0})$$ where each element of $\mathbf{H}$ (called the Hessian) is simply a second derivative evaluated at $\mathbf{x=x_0}$: $$H_{ij}(\mathbf{x_0}) = \left. \frac{\partial^2 f}{\partial x_i \partial x_j} \right|_{\mathbf{x=x_0}}$$ If the quadratic form is positive definite (all eigenvalues are positive), then $\mathbf{x=x_0}$ is a local minima, if it is negative definite (all eigenvalues are negative) then it is a local maxima. They show up in more places as well. So be rest assured that learning quadratic forms is useful! jason
 Recognitions: Science Advisor Another example is that of the intersection form on even-dimensional manifolds. On some subset of 4-manifolds ( simply-connected , I think) they determine the manifold up to homeomorphism. The properties of the intersection form of the M^4 tell a lot about the manifold itself.

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special relativity and general relativity use quadratic forms (pseudo-riemmanian metrics)
 Only have a linear algebra background so cannot understand the applications of quadratic form to calculus. Are any simple applications which can be appreciated without the use of calculus.

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