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Composition has closure and is associative, since matrix multiplication is associative. Notice that both Ref(θ) and Rot(θ) have been represented with orthogonal matrices. These matrices all have a determinant whose absolute value is unity. Rotation matrices have a determinant of +1, and reflection matrices have a determinant of −1.
Point Q is the reflection of point P through the line AB. In a plane (or, respectively, 3-dimensional) geometry, to find the reflection of a point drop a perpendicular from the point to the line (plane) used for reflection, and extend it the same distance on the other side. To find the reflection of a figure, reflect each point in the figure.
In linear algebra, a rotation matrix is a transformation matrix that is used to perform a ... this is not a proper rotation matrix; it is a reflection across the line ...
A reflection about a line or plane that does not go through the origin is not a linear transformation — it is an affine transformation — as a 4×4 affine transformation matrix, it can be expressed as follows (assuming the normal is a unit vector): [′ ′ ′] = [] [] where = for some point on the plane, or equivalently, + + + =.
In linear algebra, a Householder transformation (also known as a Householder reflection or elementary reflector) is a linear transformation that describes a reflection about a plane or hyperplane containing the origin. The Householder transformation was used in a 1958 paper by Alston Scott Householder. [1]
A linear isometry also necessarily preserves angles, therefore a linear isometry transformation is a conformal linear transformation. Examples. A linear map from to itself is an isometry (for the dot product) if and only if its matrix is unitary. [10] [11] [12] [13]
Then, interpreting both unitary matrices as well as the diagonal matrix, summarized here as , as a linear transformation of the space , the matrices and represent rotations or reflection of the space, while represents the scaling of each coordinate by the factor .
On the other hand, reflection groups are concrete, in the sense that each of its elements is the composite of finitely many geometric reflections about linear hyperplanes in some euclidean space. Technically, a reflection group is a subgroup of a linear group (or various generalizations) generated by orthogonal matrices of determinant -1.
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