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ref en français
A real matrix is '''orthogonal''' (or, more precisely, '''orthonormal''') when it has an inverse equal to its transpose:
A real matrix is '''orthogonal''' (or, more precisely, '''orthonormal''') when it has an inverse equal to its transpose<ref name="frtext">[http://benhur.teluq.uqam.ca/SPIP/inf6104/article.php3?id_article=13 Vecteurs et matrices], at http://benhur.teluq.uqam.ca, in French</ref>:
:''P''<sup>T</sup>=''P''<sup>-1</sup>
:''P''<sup>T</sup>=''P''<sup>-1</sup>
Each orthonormal matrix represents one orthonormal basis, and reciprocally:
Each orthonormal matrix represents one orthonormal basis, and reciprocally:
* If the columns of an orthonormal matrix are taken to be a set of vectors, then this set is an orthonormal basis
* If the columns of an orthonormal matrix are taken to be a set of vectors, then this set is an orthonormal basis
* If an orthonormal basis is written as the columns of a matrix, then this matrix is orthonormal
* If an orthonormal basis is written as the columns of a matrix, then this matrix is orthonormal <ref name="frtext" />
* These properties are also valid for ''lines'' instead of ''columns''
* These properties are also valid for ''lines'' instead of ''columns''
The concept generalizes to complex matrices as [[unitary matrices]]; however, in this case, instead of the the transpose it's necessary to use the conjugate-transpose.
The concept generalizes to complex matrices as [[unitary matrices]]; however, in this case, instead of the the transpose it's necessary to use the conjugate-transpose.
== References ==
<references />
[[Category:linear algebra]]
[[Category:linear algebra]]