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Line 1: − An '''orthogonal matrix''' has an inverse equal to its transpose:+ + + + + + + + + +
Should be orthonormal, not orthogonal + a few more lines + no sources!!! that's sad
A real matrix is '''orthogonal''' (or, more precisely, '''orthonormal''') when it has an inverse equal to its transpose:
:''P''<sup>T</sup>=''P''<sup>-1</sup>
:''P''<sup>T</sup>=''P''<sup>-1</sup>
The term comes from the fact that the canonical [[orthonormal basis]] of the <math>\mathbb{R}^n\,</math> 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
* 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.
[[Category:linear algebra]]
[[Category:linear algebra]]