Editing Orthogonal matrix (section)

===Group properties===
The inverse of every orthogonal matrix is again orthogonal, as is the matrix product of two orthogonal matrices. In fact, the set of all {{math|''n'' × ''n''}} orthogonal matrices satisfies all the axioms of a [[group (mathematics)|group]]. It is a [[compact space|compact]] [[Lie group]] of dimension {{math|{{sfrac|''n''(''n'' − 1)|2}}}}, called the [[orthogonal group]] and denoted by {{math|O(''n'')}}.

The orthogonal matrices whose determinant is +1 form a [[connected space|path-connected]] [[normal subgroup]] of {{math|O(''n'')}} of [[index of a subgroup|index]] 2, the [[special orthogonal group]] {{math|SO(''n'')}} of rotations. The [[quotient group]] {{math|O(''n'')/SO(''n'')}} is isomorphic to {{math|O(1)}}, with the projection map choosing [+1] or [−1] according to the determinant. Orthogonal matrices with determinant −1 do not include the identity, and so do not form a subgroup but only a [[coset]]; it is also (separately) connected. Thus each orthogonal group falls into two pieces; and because the projection map [[exact sequence|splits]], {{math|O(''n'')}} is a [[semidirect product]] of {{math|SO(''n'')}} by {{math|O(1)}}. In practical terms, a comparable statement is that any orthogonal matrix can be produced by taking a rotation matrix and possibly negating one of its columns, as we saw with {{nowrap|2 × 2}} matrices. If {{mvar|n}} is odd, then the semidirect product is in fact a [[direct product of groups|direct product]], and any orthogonal matrix can be produced by taking a rotation matrix and possibly negating all of its columns. This follows from the property of determinants that negating a column negates the determinant, and thus negating an odd (but not even) number of columns negates the determinant.

Now consider {{math|(''n'' + 1) × (''n'' + 1)}} orthogonal matrices with bottom right entry equal to 1. The remainder of the last column (and last row) must be zeros, and the product of any two such matrices has the same form. The rest of the matrix is an {{math|''n'' × ''n''}} orthogonal matrix; thus {{math|O(''n'')}} is a subgroup of {{math|O(''n'' + 1)}} (and of all higher groups).

<math display="block">\begin{bmatrix}
  & & & 0\\
  & \mathrm{O}(n) & & \vdots\\
  & & & 0\\
  0 & \cdots & 0 & 1
\end{bmatrix}</math>

Since an elementary reflection in the form of a [[Householder matrix]] can reduce any orthogonal matrix to this constrained form, a series of such reflections can bring any orthogonal matrix to the identity; thus an orthogonal group is a [[reflection group]]. The last column can be fixed to any unit vector, and each choice gives a different copy of {{math|O(''n'')}} in {{math|O(''n'' + 1)}}; in this way {{math|O(''n'' + 1)}} is a [[fiber bundle|bundle]] over the unit sphere {{math|''S''<sup>''n''</sup>}} with fiber {{math|O(''n'')}}.

Similarly, {{math|SO(''n'')}} is a subgroup of {{math|SO(''n'' + 1)}}; and any special orthogonal matrix can be generated by [[Givens rotation|Givens plane rotations]] using an analogous procedure. The bundle structure persists: {{math|SO(''n'') ↪ SO(''n'' + 1) → ''S''<sup>''n''</sup>}}. A single rotation can produce a zero in the first row of the last column, and series of {{math|''n'' − 1}} rotations will zero all but the last row of the last column of an {{math|''n'' × ''n''}} rotation matrix. Since the planes are fixed, each rotation has only one degree of freedom, its angle. By induction, {{math|SO(''n'')}} therefore has
<math display="block">(n-1) + (n-2) + \cdots + 1 = \frac{n(n-1)}{2}</math>
degrees of freedom, and so does {{math|O(''n'')}}.

Permutation matrices are simpler still; they form, not a Lie group, but only a finite group, the order [[factorial|{{math|''n''!}}]] [[symmetric group]] {{math|S<sub>''n''</sub>}}. By the same kind of argument, {{math|S<sub>''n''</sub>}} is a subgroup of {{math|S<sub>''n'' + 1</sub>}}. The even permutations produce the subgroup of permutation matrices of determinant +1, the order {{math|{{sfrac|''n''!|2}}}} [[alternating group]].