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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, + + + =.
Noting that any identity matrix is a rotation matrix, and that matrix multiplication is associative, we may summarize all these properties by saying that the n × n rotation matrices form a group, which for n > 2 is non-abelian, called a special orthogonal group, and denoted by SO(n), SO(n,R), SO n, or SO n (R), the group of n × n rotation ...
It turns out that g ∈ SO(3) represented in this way by Π u (g) can be expressed as a matrix Π u (g) ∈ SU(2) (where the notation is recycled to use the same name for the matrix as for the transformation of it represents). To identify this matrix, consider first a rotation g φ about the z-axis through an angle φ,
The most external matrix rotates the other two, leaving the second rotation matrix over the line of nodes, and the third one in a frame comoving with the body. There are 3 × 3 × 3 = 27 possible combinations of three basic rotations but only 3 × 2 × 2 = 12 of them can be used for representing arbitrary 3D rotations as Euler angles.
An infinitesimal rotation matrix or differential rotation matrix is a matrix representing an infinitely small rotation.. While a rotation matrix is an orthogonal matrix = representing an element of () (the special orthogonal group), the differential of a rotation is a skew-symmetric matrix = in the tangent space (the special orthogonal Lie algebra), which is not itself a rotation matrix.
From linear algebra one knows that a certain matrix can be represented in another basis through the transformation ′ = where is the basis transformation matrix. If the vectors b {\displaystyle b} respectively c {\displaystyle c} are the z-axis in one basis respectively another, they are perpendicular to the y-axis with a certain angle t ...
is the rotation matrix through an angle θ counterclockwise about the axis k, and I the 3 × 3 identity matrix. [4] This matrix R is an element of the rotation group SO(3) of ℝ 3 , and K is an element of the Lie algebra s o ( 3 ) {\displaystyle {\mathfrak {so}}(3)} generating that Lie group (note that K is skew-symmetric, which characterizes ...
Writing the general matrix transformation of coordinates as the matrix equation [′ ′ ′ ′] = [′ ′ ′ ′] [′ ′ ′ ′] allows the transformation of other physical quantities that cannot be expressed as four-vectors; e.g., tensors or spinors of any order in 4d spacetime, to be defined.