1. Feb 18, 2013

### matqkks

What are the real life applications of quadratic forms? I have used them to sketch conics but are there any other applications?

2. Feb 18, 2013

### jasonRF

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

Last edited: Feb 18, 2013
3. Feb 18, 2013

### Bacle2

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

4. Feb 18, 2013

### robphy

special relativity and general relativity use quadratic forms (pseudo-riemmanian metrics)

5. Feb 23, 2013

### matqkks

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.

6. Feb 23, 2013

### jbunniii

One way quadratic forms are used in linear algebra is to define the norm of a matrix.

Recall that the norm (length) of a vector in $\mathbb{R}^N$ is simply $\|v\| = \sqrt{v^T v}$. If $v = (a_1, a_2, \ldots, a_N)$, then $\|v\|$ can be written as $\sqrt{a_1^2 + a_2^2 + \ldots + a_N^2}$.

If $A$ is an $N \times N$ matrix, then we may define a norm for $A$ as follows: $\|A\| = \max \|Ax\|$, where the max is taken over all unit vectors $x$, i.e. all vectors with $\|x\| = 1$. Note that $\|Ax\| = \sqrt{x^T (A^T A) x}$, so the norm is based upon the quadratic form $x^T(A^T A) x$.

Note that in general, $A$ maps the unit sphere into an ellipsoid. The norm of $A$ is telling us the distance from the origin to the ellipsoid along its longest axis. This turns out to be very useful in numerical linear algebra and matrix analysis.