Quadratic Forms: Beyond Sketching Conics

[matqkks]

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

 

[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

 

[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
about the manifold itself.

 

[robphy]

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

 

[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.

 

[jbunniii]

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.

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.