Difference between revisions of "Orthogonal matrix"
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(Should be orthonormal, not orthogonal + a few more lines + no sources!!! that's sad) |
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| − | 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> | ||
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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]] | ||
Revision as of 14:33, May 13, 2010
A real matrix is orthogonal (or, more precisely, orthonormal) when it has an inverse equal to its transpose[1]:
- PT=P-1
The term comes from the fact that the canonical orthonormal basis of the is transformed by any orthonormal matrix (and only by orthonormal matrices) into another orthonormal basis.
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 an orthonormal basis is written as the columns of a matrix, then this matrix is orthonormal [1]
- 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.
References
- ↑ 1.0 1.1 Vecteurs et matrices, at http://benhur.teluq.uqam.ca, in French