rotaatioMatriisia

Rotaatiomatriisi (rotation matrix) is a square matrix that represents a rotation of vectors in n-dimensional Euclidean space. In two dimensions, a rotation by angle θ is represented by R(θ) = [[cos θ, -sin θ], [sin θ, cos θ]]. In three dimensions, a rotation is described by an axis u (a unit vector) and an angle θ; one common representation is Rodrigues' formula: R = I cos θ + (1 − cos θ) u u^T + sin θ [u]_×, where [u]_× is the cross-product matrix of u.

Properties: Rotation matrices are orthogonal: R^T R = I, and det(R) = 1. They preserve the Euclidean inner product and vector lengths, so |Rv| = |v| and (Rv) · (Rw) = v · w. The set of all n×n rotation matrices forms the special orthogonal group SO(n).

Eigenstructure: For 2D rotations, eigenvalues are e^(iθ) and e^(−iθ) unless θ = 0 or π. In 3D, a rotation about axis u leaves vectors along u unchanged (eigenvalue 1); the other directions have complex eigenvalues on the unit circle.

Applications and notes: Rotation matrices are used for coordinate transformations in computer graphics, robotics, aerospace, and physics. In numerical work, finite precision can introduce slight departures from orthogonality; re-orthogonalization or projection back to SO(n) may be employed. In Finnish contexts, rotaatiomatriisi or rotaatiomatriisia refers to the same concept.