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Definition/Summary
A set of vectors
<br /> \left\{\mathbf{v}^{(i)}\right\}<br />
is called "orthonormal" if the vectors of the set are normalized to 1 and are orthogonal to each other.
<br /> \mathbf{v}^{(i)}\cdot\mathbf{v}^{(j)}=\delta_{ij}\;,<br />
where \delta_{ij} 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
<br /> \mathbf{f}\cdot\mathbf{g}\equiv \int_{-\infty}^{\infty} w(x) f^*(x) g(x)dx\;,<br />
where w(x) is an appropriate weighing function. An example where w(x) is a unitstep function on the interval 2p, and where f and g are trig functions is given below.
In what follow, the constants m and n are nonnegative real integers. The orthogonality properties of the trigonometric system are expressed by:
<br /> \begin{align*}<br /> \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} \\<br /> \int_{ - p + x_0 }^{p + x_0 } {\cos {\frac{m\pi x}{p}} \cos {\frac{n\pi x}{p}} dx} &= <br /> \left\{<br /> \begin{array}{cll}<br /> 2p & \text{for}&m=n=0\\<br /> p & \text{for} &m=n>0\\<br /> 0 & \text{for} &m\neq n<br /> \end{array}<br /> \right. \\<br /> \int_{ - p + x_0 }^{p + x_0 } {\sin {\frac{m\pi x}{p}} \sin {\frac{n\pi x}{p}} d x} &= <br /> \left\{<br /> \begin{array}{cll}<br /> 0 & \text{for}&m=n=0\\<br /> p & \text{for} &m=n>0\\<br /> 0 & \text{for} &m\neq n.<br /> \end{array}<br /> \right.<br /> \end{align*}<br />
Here 2p is the period, and x_0 is an arbitrary constant. We are allowed to add the constant x_0 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!
A set of vectors
<br /> \left\{\mathbf{v}^{(i)}\right\}<br />
is called "orthonormal" if the vectors of the set are normalized to 1 and are orthogonal to each other.
<br /> \mathbf{v}^{(i)}\cdot\mathbf{v}^{(j)}=\delta_{ij}\;,<br />
where \delta_{ij} 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
<br /> \mathbf{f}\cdot\mathbf{g}\equiv \int_{-\infty}^{\infty} w(x) f^*(x) g(x)dx\;,<br />
where w(x) is an appropriate weighing function. An example where w(x) is a unitstep function on the interval 2p, and where f and g are trig functions is given below.
In what follow, the constants m and n are nonnegative real integers. The orthogonality properties of the trigonometric system are expressed by:
<br /> \begin{align*}<br /> \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} \\<br /> \int_{ - p + x_0 }^{p + x_0 } {\cos {\frac{m\pi x}{p}} \cos {\frac{n\pi x}{p}} dx} &= <br /> \left\{<br /> \begin{array}{cll}<br /> 2p & \text{for}&m=n=0\\<br /> p & \text{for} &m=n>0\\<br /> 0 & \text{for} &m\neq n<br /> \end{array}<br /> \right. \\<br /> \int_{ - p + x_0 }^{p + x_0 } {\sin {\frac{m\pi x}{p}} \sin {\frac{n\pi x}{p}} d x} &= <br /> \left\{<br /> \begin{array}{cll}<br /> 0 & \text{for}&m=n=0\\<br /> p & \text{for} &m=n>0\\<br /> 0 & \text{for} &m\neq n.<br /> \end{array}<br /> \right.<br /> \end{align*}<br />
Here 2p is the period, and x_0 is an arbitrary constant. We are allowed to add the constant x_0 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!