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


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.


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


The mapping defined by an orthogonal matrix with respect to the standard basis, , , preserves the standard inner product and hence defines an orthogonal mapping. More generally, if and are inner product spaces with inner products , , then a linear mapping such that is called an orthogonal mapping.

Any non-singular (complex or real) matrix admits a polar decomposition with and symmetric and and orthogonal.


[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)
How to Cite This Entry:
Orthogonal matrix. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by D.A. Suprunenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article