  ## Encyclopedia > Orthonormal matrix

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# 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

[itex]G^*G = I_k [/itex]

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

If the n-by-k matrix G is orthonormal, then kn. 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.

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