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What is orthonormal

  1. Jul 23, 2014 #1
    Definition/Summary

    A set of vectors
    [tex]
    \left\{\mathbf{v}^{(i)}\right\}
    [/tex]
    is called "orthonormal" if the vectors of the set are normalized to 1 and are orthogonal to each other.

    [tex]
    \mathbf{v}^{(i)}\cdot\mathbf{v}^{(j)}=\delta_{ij}\;,
    [/tex]
    where [itex]\delta_{ij}[/itex] is the Kronecker delta function.

    Equations



    Extended explanation

    Functions may also be considered as vectors with an appropriately defined dot-product. For example, the dot product for functions of a single variable could be defined as
    [tex]
    \mathbf{f}\cdot\mathbf{g}\equiv \int_{-\infty}^{\infty} w(x) f^*(x) g(x)dx\;,
    [/tex]
    where [itex]w(x)[/itex] is an appropriate weighing function. An example where [itex]w(x)[/itex] is a unitstep function on the interval 2p, and where f and g are trig functions is given below.

    In what follow, the constants [itex]m[/itex] and [itex]n[/itex] are nonnegative real integers. The orthogonality properties of the trigonometric system are expressed by:

    [tex]
    \begin{align*}
    \int_{ - p + x_0 }^{p + x_0 } {\sin {\frac{m\pi x}{p}}\cos {\frac{n\pi x}{p}} x} &= 0 \quad \text{for all m and n} \\
    \int_{ - p + x_0 }^{p + x_0 } {\cos {\frac{m\pi x}{p}} \cos {\frac{n\pi x}{p}} dx} &=
    \left\{
    \begin{array}{cll}
    2p & \text{for}&m=n=0\\
    p & \text{for} &m=n>0\\
    0 & \text{for} &m\neq n
    \end{array}
    \right. \\
    \int_{ - p + x_0 }^{p + x_0 } {\sin {\frac{m\pi x}{p}} \sin {\frac{n\pi x}{p}} d x} &=
    \left\{
    \begin{array}{cll}
    0 & \text{for}&m=n=0\\
    p & \text{for} &m=n>0\\
    0 & \text{for} &m\neq n.
    \end{array}
    \right.
    \end{align*}
    [/tex]

    Here [itex]2p[/itex] is the period, and [itex]x_0[/itex] is an arbitrary constant. We are allowed to add the constant [itex]x_0[/itex] to the limits, since we are integrating over a full period.

    * This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!
     
  2. jcsd
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