# 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.

How to Cite This Entry:
Orthogonal matrix. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Orthogonal_matrix&oldid=49505
This article was adapted from an original article by D.A. Suprunenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article