Editing Orthogonal matrix (section)

===Primitives===
The most elementary permutation is a transposition, obtained from the identity matrix by exchanging two rows. Any {{math|''n'' × ''n''}} permutation matrix can be constructed as a product of no more than {{math|''n'' − 1}} transpositions.

A [[Householder reflection]] is constructed from a non-null vector {{math|'''v'''}} as
<math display="block">Q = I - 2 \frac{{\mathbf v}{\mathbf v}^\mathrm{T}}{{\mathbf v}^\mathrm{T}{\mathbf v}} .</math>

Here the numerator is a symmetric matrix while the denominator is a number, the squared magnitude of {{math|'''v'''}}. This is a reflection in the hyperplane perpendicular to {{math|'''v'''}} (negating any vector component parallel to {{math|'''v'''}}). If {{math|'''v'''}} is a unit vector, then {{math|1=''Q'' = ''I'' − 2'''vv'''<sup>T</sup>}} suffices. A Householder reflection is typically used to simultaneously zero the lower part of a column. Any orthogonal matrix of size {{nowrap|''n'' × ''n''}} can be constructed as a product of at most {{mvar|n}} such reflections.

A [[Givens rotation]] acts on a two-dimensional (planar) subspace spanned by two coordinate axes, rotating by a chosen angle. It is typically used to zero a single subdiagonal entry. Any rotation matrix of size {{math|''n'' × ''n''}} can be constructed as a product of at most {{math|{{sfrac|''n''(''n'' − 1)|2}}}} such rotations. In the case of {{nowrap|3 × 3}} matrices, three such rotations suffice; and by fixing the sequence we can thus describe all {{nowrap|3 × 3}} rotation matrices (though not uniquely) in terms of the three angles used, often called [[Euler angles]].

A [[Jacobi rotation]] has the same form as a Givens rotation, but is used to zero both off-diagonal entries of a {{nowrap|2 × 2}} symmetric submatrix.