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Quadratic Forms

  1. Feb 18, 2013 #1
    What are the real life applications of quadratic forms? I have used them to sketch conics but are there any other applications?
     
  2. jcsd
  3. Feb 18, 2013 #2

    jasonRF

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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
    [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 twice-differentiable 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{x-x_0})^T \mathbf{H(\mathbf{x_0})}(\mathbf{x-x_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
     
    Last edited: Feb 18, 2013
  4. Feb 18, 2013 #3

    Bacle2

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    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.
     
  5. Feb 18, 2013 #4

    robphy

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    special relativity and general relativity use quadratic forms (pseudo-riemmanian metrics)
     
  6. Feb 23, 2013 #5
    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.
     
  7. Feb 23, 2013 #6

    jbunniii

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    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.
     
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