Editing Orthogonal matrix (section)

==Overview==
[[File:Matrix multiplication transpose.svg|thumb|275px|Visual understanding of multiplication by the transpose of a matrix. If A is an orthogonal matrix and B is its transpose, the ij-th element of the product AA<sup>T</sup> will vanish if i≠j, because the i-th row of A is orthogonal to the j-th row of A.]]

An orthogonal matrix is the real specialization of a unitary matrix, and thus always a [[normal matrix]]. Although we consider only real matrices here, the definition can be used for matrices with entries from any [[field (mathematics)|field]]. However, orthogonal matrices arise naturally from [[dot product]]s, and for matrices of complex numbers that leads instead to the unitary requirement. Orthogonal matrices preserve the dot product,<ref>[http://tutorial.math.lamar.edu/Classes/LinAlg/OrthogonalMatrix.aspx "Paul's online math notes"]{{Full citation needed|date=January 2013|note=See talk page.}}, Paul Dawkins, [[Lamar University]], 2008. Theorem 3(c)</ref> so, for vectors {{math|'''u'''}} and {{math|'''v'''}} in an {{mvar|n}}-dimensional real [[Euclidean space]]
<math display="block">{\mathbf u} \cdot {\mathbf v} = \left(Q {\mathbf u}\right) \cdot \left(Q {\mathbf v}\right) </math>
where {{mvar|Q}} is an orthogonal matrix. To see the inner product connection, consider a vector {{math|'''v'''}} in an {{mvar|n}}-dimensional real [[Euclidean space]]. Written with respect to an orthonormal basis, the squared length of {{math|'''v'''}} is {{math|'''v'''<sup>T</sup>'''v'''}}. If a linear transformation, in matrix form {{math|''Q'''''v'''}}, preserves vector lengths, then
<math display="block">{\mathbf v}^\mathrm{T}{\mathbf v} = (Q{\mathbf v})^\mathrm{T}(Q{\mathbf v}) = {\mathbf v}^\mathrm{T} Q^\mathrm{T} Q {\mathbf v} .</math>

Thus [[dimension (vector space)|finite-dimensional]] linear isometries—rotations, reflections, and their combinations—produce orthogonal matrices. The converse is also true: orthogonal matrices imply orthogonal transformations. However, linear algebra includes orthogonal transformations between spaces which may be neither finite-dimensional nor of the same dimension, and these have no orthogonal matrix equivalent.

Orthogonal matrices are important for a number of reasons, both theoretical and practical. The {{math|''n'' × ''n''}} orthogonal matrices form a [[group (mathematics)|group]] under matrix multiplication, the [[orthogonal group]] denoted by {{math|O(''n'')}}, which—with its subgroups—is widely used in mathematics and the physical sciences. For example, the [[point group]] of a molecule is a subgroup of O(3). Because floating point versions of orthogonal matrices have advantageous properties, they are key to many algorithms in numerical linear algebra, such as [[QR decomposition|{{mvar|QR}} decomposition]]. As another example, with appropriate normalization the [[discrete cosine transform]] (used in [[MP3]] compression) is represented by an orthogonal matrix.