In
linear algebra, an orthogonal transformation is a
linear transformation ''T'' : ''V'' → ''V'' on a
real
Real may refer to:
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* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010) ...
inner product space ''V'', that preserves the
inner product. That is, for each pair of elements of ''V'', we have
:
Since the lengths of vectors and the angles between them are defined through the inner product, orthogonal transformations preserve lengths of vectors and angles between them. In particular, orthogonal transformations map
orthonormal bases to orthonormal bases.
Orthogonal transformations are
injective: if
then
, hence
, so the
kernel
Kernel may refer to:
Computing
* Kernel (operating system), the central component of most operating systems
* Kernel (image processing), a matrix used for image convolution
* Compute kernel, in GPGPU programming
* Kernel method, in machine learni ...
of
is trivial.
Orthogonal transformations in two- or three-
dimensional
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordi ...
Euclidean space are stiff
rotations,
reflections, or combinations of a rotation and a reflection (also known as
improper rotations). Reflections are transformations that reverse the direction front to back, orthogonal to the mirror plane, like (real-world) mirrors do. The
matrices corresponding to proper rotations (without reflection) have a
determinant of +1. Transformations with reflection are represented by matrices with a determinant of −1. This allows the concept of rotation and reflection to be generalized to higher dimensions.
In finite-dimensional spaces, the matrix representation (with respect to an
orthonormal basis) of an orthogonal transformation is an
orthogonal matrix
In linear algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors.
One way to express this is
Q^\mathrm Q = Q Q^\mathrm = I,
where is the transpose of and is the identity ma ...
. Its rows are mutually orthogonal vectors with unit norm, so that the rows constitute an orthonormal basis of ''V''. The columns of the matrix form another orthonormal basis of ''V''.
If an orthogonal transformation is
invertible (which is always the case when ''V'' is finite-dimensional) then its inverse is another orthogonal transformation. Its matrix representation is the transpose of the matrix representation of the original transformation.
Examples
Consider the inner-product space
with the standard euclidean inner product and standard basis. Then, the matrix transformation
:
is orthogonal. To see this, consider
:
Then,
:
The previous example can be extended to construct all orthogonal transformations. For example, the following matrices define orthogonal transformations on
:
:
See also
*
Improper rotation
*
Linear transformation
*
Orthogonal matrix
In linear algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors.
One way to express this is
Q^\mathrm Q = Q Q^\mathrm = I,
where is the transpose of and is the identity ma ...
*
Unitary transformation
In mathematics, a unitary transformation is a transformation that preserves the inner product: the inner product of two vectors before the transformation is equal to their inner product after the transformation.
Formal definition
More precisely, ...
References
{{Reflist
Linear algebra