orthonormal set

Definition

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

More verbosely, we may say that an orthonormal setis a subset $S$ of an inner product spacesuch that the following two conditions hold:

1. If $x,y\in S$ and $x\ne y$, then $x$ is orthogonal (http://planetmath.org/OrthogonalVector) 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 normin 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.