Orthogonal matrix

A matrix over a commutative ring $ R $ with identity $ 1 $ for which the 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 $ \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

$$ a _ {j} = \left \| \begin{array}{rc} \cos \phi _ {j} &\sin \phi _ {j} \\ - \sin \phi _ {j} &\cos \phi _ {j} \\ \end{array} \right \| . $$

A non-singular complex matrix $ a $ is similar to a complex orthogonal matrix if and only if its system of 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

[1]	A.I. Mal'tsev, "Foundations of linear algebra" , Freeman (1963) (Translated from Russian)

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 $ M = SQ = Q _ {1} S _ {1} $ with $ S $ and $ S _ {1} $ symmetric and $ Q $ and $ Q _ {1} $ orthogonal.

References

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