Encyclopedia > Orthonormal matrix

  Article Content

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

<math>G^*G = I_k </math>

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.



All Wikipedia text is available under the terms of the GNU Free Documentation License

 
  Search Encyclopedia

Search over one million articles, find something about almost anything!
 
 
  
  Featured Article
242

... - 4th century Decades: 190s 200s 210s 220s 230s - 240s - 250s 260s 270s 280s 290s Years: 237 238 239 240 241 - 242 - 243 244 245 246 247 Events Patriarch ...

 
 
 
This page was created in 99.7 ms