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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.

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