What is Orthonormal: Definition & Explanation

[Greg Bernhardt]

Definition/Summary

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

$$\mathbf{v}^{(i)}\cdot\mathbf{v}^{(j)}=\delta_{ij}\;,$$
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
$$\mathbf{f}\cdot\mathbf{g}\equiv \int_{-\infty}^{\infty} w(x) f^*(x) g(x)dx\;,$$
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:

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

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!

 

[fresh_42]

See also: https://www.physicsforums.com/insights/hilbert-spaces-relatives/ (2 parts)