Orthonormal set

A set of vectors is called a orthonormal set if all vectors are orthogonal to every vector in the set and are all unit vectors. Mathematically, this is stated as

$\langle\mathbf{v}_i\cdot\mathbf{v}_j\rangle=\delta_{ij}=\begin{cases}1&i=j\\ 0&i\ne j\end{cases}$

where $\delta_{ij}$ is the Kronecker delta function and $\langle\mathbf{v}_i \cdot\mathbf{v}_j\rangle$ is the inner product (which can also be extended to real functions; as such a set of functions can be orthonormal). The span of the set forms an orthonormal basis. An example of this is Cartesian coordinates.

A matrix whose columns form an orthonormal set is called an orthogonal matrix, and will have the property

$A^{-1}=A^\mathrm{T}$

where $A^{-1}$ is the inverse of $A$ and $A^T$ is its transpose. Real orthogonal matrices will have a determinant of either 1 or -1. An extension to complex matrices is a matrix whose inverse is equal to its conjugate transpose, and is called a unitary matrix.