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A [[Matrix|matrix]] over a commutative ring  $  R $
with identity  $  1 $
for which the [[Transposed matrix|transposed matrix]] coincides with the inverse. The determinant of an orthogonal matrix is equal to  $  \pm  1 $. 
The set of all orthogonal matrices of order  $  n $
over  $  R $
forms a subgroup of the [[General linear group|general linear group]]  $   \mathop{\rm GL} _ {n} ( R) $. 
For any real orthogonal matrix  $  a $
there is a real orthogonal matrix  $  c $
such that

$$ 
cac  ^ {-} 1  =   \mathop{\rm diag}  [\pm  1 \dots \pm  1 , a _ {1} \dots a _ {t} ],
$$

where

$$ 
a _ {j}  =  \left \|

A non-singular complex matrix  $  a $
is similar to a complex orthogonal matrix if and only if its system of [[Elementary divisors|elementary divisors]] possesses the following properties:

1) for  $  \lambda \neq \pm  1 $, 
the elementary divisors  $  ( x - \lambda )  ^ {m} $
and  $  ( x - \lambda  ^ {-} 1 )  ^ {m} $
are repeated the same number of times;

2) each elementary divisor of the form  $  ( x \pm  1)  ^ {2l} $
is repeated an even number of times.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.I. Mal'tsev,   "Foundations of linear algebra" , Freeman  (1963)  (Translated from Russian)</TD></TR></table>

====Comments====
The mapping  $  \alpha :  \mathbf R  ^ {n} \rightarrow \mathbf R  ^ {n} $
defined by an orthogonal matrix  $  A $
with respect to the standard basis,  $  \alpha ( x) = Ax $, 
$  x \in \mathbf R  ^ {n} $, 
preserves the standard inner product and hence defines an orthogonal mapping. More generally, if  $  V $
and  $  W $
are inner product spaces with inner products  $  \langle   , \rangle _ {V} $, 
$  \langle   , \rangle _ {W} $, 
then a linear mapping  $  \alpha :  V \rightarrow W $
such that  $  \langle  \alpha ( x) , \alpha ( y) \rangle _ {W} = \langle  x, y \rangle _ {V} $
is called an orthogonal mapping.

Any non-singular (complex or real) matrix  $  M $
admits a [[Polar decomposition|polar decomposition]]  $  M = SQ = Q _ {1} S _ {1} $
with  $  S $
and  $  S _ {1} $
symmetric and  $  Q $
and  $  Q _ {1} $
orthogonal.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  F.R. [F.R. Gantmakher] Gantmacher,   "The theory of matrices" , '''1''' , Chelsea, reprint  (1959)  pp. 263ff  (Translated from Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  W. Noll,   "Finite dimensional spaces" , M. Nijhoff  (1987)  pp. Sect. 43</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  H.W. Turnball,   A.C. Aitken,   "An introduction to the theory of canonical matrices" , Blackie &amp; Son  (1932)</TD></TR></table>