Orthonormal matrix

An orthonormal matrix is a matrix whose column vectors are orthonormal. In other words, if G is an n-by-k orthonormal matrix then G^*G=I_n, 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.