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) |
Orthogonal matrix. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Orthogonal_matrix&oldid=49346