# 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)

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.