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.
|||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)|
Orthogonal matrix. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Orthogonal_matrix&oldid=17418