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Quadratic Forms 
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#1
Feb1813, 06:29 AM

P: 153

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



#2
Feb1813, 09:41 AM

P: 698

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
[tex] T = \frac{1}{2}\mathbf{\omega^T I \omega} [/tex] where [itex] \mathbf{\omega}[/itex] is the angular velocity vector (3x1) and [itex]\mathbf{I}[/itex] 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 [itex]\mathbf{I}[/itex] 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 twicedifferentiable function of N variables [itex]f(\mathbf{x})[/itex], where [itex] \mathbf{x}[/itex] is the Nx1 vector of variables. If we want to find a local maximum and minimum, these will occur at a location (call it [itex]\mathbf{x=x_0}[/itex]) where the first derivatives are zero, [tex] \left. \nabla f(x) \right_{\mathbf{x=x_0}} = \mathbf{0}. [/tex] This equation represents N scalar equations. The first two terms of the Taylor expansion of f about [itex]\mathbf{x=x_0}[/itex] is then [tex] f(\mathbf{x}) \approx f(\mathbf{x_0}) + \frac{1}{2}(\mathbf{xx_0})^T \mathbf{H(\mathbf{x_0})}(\mathbf{xx_0}) [/tex] where each element of [itex]\mathbf{H}[/itex] (called the Hessian) is simply a second derivative evaluated at [itex]\mathbf{x=x_0}[/itex]: [tex] H_{ij}(\mathbf{x_0}) = \left. \frac{\partial^2 f}{\partial x_i \partial x_j} \right_{\mathbf{x=x_0}} [/tex] If the quadratic form is positive definite (all eigenvalues are positive), then [itex]\mathbf{x=x_0}[/itex] 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 


#3
Feb1813, 03:31 PM

Sci Advisor
P: 1,172

Another example is that of the intersection form on evendimensional manifolds. On some subset of 4manifolds ( simplyconnected , 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. 


#4
Feb1813, 03:41 PM

Sci Advisor
HW Helper
PF Gold
P: 4,139

Quadratic Forms
special relativity and general relativity use quadratic forms (pseudoriemmanian metrics)



#5
Feb2313, 01:36 PM

P: 153

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
Feb2313, 05:36 PM

Sci Advisor
HW Helper
PF Gold
P: 3,289

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. 


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