# Orthogonal matrix

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.

#### References

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

#### Comments

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.

#### 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) |

**How to Cite This Entry:**

Orthogonal matrix.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Orthogonal_matrix&oldid=17418