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.

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

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

where I_n 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.