Orthonormal matrix

In linear algebra, an orthonormal matrix is a matrix whose columns, treated as vectors, are orthonormal. That is, the dot product of any two different columns is zero (or more generally, the inner product).

This means that if if G is an n-by-k orthonormal matrix, and G^T denotes its transpose, then:

$G^{T}G=I_{k}$

where I_k is the identity matrix.

Moreover, if k<n then there exists an n-by-(n-k) orthonormal matrix H such that U=(G H) is a unitary matrix.

If G is real then H can chosen to be real and U is therefore an orthogonal matrix.

Of course, unitary matrix and orthogonal matrix are orthonormal matrices.