What is orthonormal

1. Jul 23, 2014

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*} \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*}

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.

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