# Orthogonal matrix

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A matrix over a commutative ring with identity for which the transposed matrix coincides with the inverse. The determinant of an orthogonal matrix is equal to . The set of all orthogonal matrices of order over forms a subgroup of the general linear group . For any real orthogonal matrix there is a real orthogonal matrix such that where A non-singular complex matrix is similar to a complex orthogonal matrix if and only if its system of elementary divisors possesses the following properties:

1) for , the elementary divisors and are repeated the same number of times;

2) each elementary divisor of the form 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=17418
This article was adapted from an original article by D.A. Suprunenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article