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The term R-matrix has several meanings, depending on the field of study.. The term R-matrix is used in connection with the Yang–Baxter equation, first introduced in the field of statistical mechanics in the works of J. B. McGuire in 1964 [1] and C. N. Yang in 1967 [2] and in the group algebra [] of the symmetric group in the work of A. A. Jucys in 1966.
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 generating that Lie group (note that K is skew-symmetric, which characterizes ()).
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 ...
For any vector r 0, consider r(t) = A(t)r 0 and differentiate it: = = () The derivative of a vector is the linear velocity of its tip. Since A is a rotation matrix, by definition the length of r ( t ) is always equal to the length of r 0 , and hence it does not change with time.
The set M(n, R) (also denoted M n (R) [7]) of all square n-by-n matrices over R is a ring called matrix ring, isomorphic to the endomorphism ring of the left R-module R n. [58] If the ring R is commutative, that is, its multiplication is commutative, then the ring M(n, R) is also an associative algebra over R.
This is a Lie-algebraic derivation, in contrast to the geometric one in the article Rodrigues' rotation formula. [1] Due to the existence of the above-mentioned exponential map, the unit vector ω representing the rotation axis, and the angle θ are sometimes called the exponential coordinates of the rotation matrix R.
The formula for left multiplication, a special case of matrix multiplication, is: = = Any linear transformation is a continuous function (see below ). Also, a matrix defines an open map from R n to R m if and only if the rank of the matrix equals to m .
In many cases, such a matrix R can be obtained by an explicit formula. Square roots that are not the all-zeros matrix come in pairs: if R is a square root of M, then −R is also a square root of M, since (−R)(−R) = (−1)(−1)(RR) = R 2 = M. A 2×2 matrix with two distinct nonzero eigenvalues has four square roots.