Orthonormal matrix

In linear algebra, an orthonormal matrix is a (not necessarily square) matrix with real or complex entries whose columns, treated as vectors in Rⁿ or Cⁿ, are orthonormal with respect to the standard inner product of Rⁿ or Cⁿ.

This means that an n-by-k matrix G is orthonormal if and only if

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

where G^* denotes the conjugate transpose of G and I_k is the k-by-k identity matrix.

If the n-by-k matrix G is orthonormal, then k ≤ n. The real n-by-k orthonormal matrices are precisely the matrices that result from deleting n-k columns from an orthogonal matrix; the complex n-by-k orthonormal matrices are precisely the matrices that result from deleting n-k columns from an unitary matrix. In particular, unitary and orthogonal matrices are themselves orthonormal.