orthonormal set

Definition

An orthonormal set is a subset $S$ of an inner product space, such that $\ip{x,y}=\delta_{xy}$ for all $x,y\in S$ . Here $\ip{\cdot,\cdot}$ is the inner product, and $\delta$ is the Kronecker delta.

More verbosely, we may say that an orthonormal set is a subset $S$ of an inner product space such that the following two conditions hold:

1. If $x,y \in S$ and $x\ne y$ , then $x$ is orthogonal to $y$ .
2. If $x \in S$ , then the norm of $x$ is $1$ .

Stated this way, the origin of the term is clear: an orthonormal set of vectors is both orthogonal and normalized.

Notes

Note that the empty set is orthonormal, as is a set consisting of a single vector of unit norm in an inner product space.

The columns (or rows) of a real orthogonal matrix form an orthonormal set. In fact, this is an example of an orthonormal basis.

Applications

A standard application is finding an orthonormal basis for a vector space, such as by Gram-Schmidt orthonormalization. Orthonormal bases are computationally simple to work with.