The rotation matrix for 2d and 3d using solely trigonometric functions are respectively given here and here and are well known. In both, they use trigonometric functions. Is there a iterative formula, A(row, column, dimension), that gives the entries of a rotation matrix in arbitrary dimensions (using trigonometric functions, not quaternions or any other things.)?
2026-03-30 12:00:25.1774872025
Formula for Entries of Rotation Matrix in k-Dimensions
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According to your first reference, rotation matrices can be characterized as orthogonal matrices with determinant 1; that is, a square matrix $R$ is a rotation matrix if and only if $R^T=R^{-1}$ and $\det R = 1$. It is known, that the effect of any ($n$-dimensional) orthogonal matrix separates into independent actions on orthogonal two-dimensional subspaces. That is, if $Q$ is special orthogonal then one can always find an orthogonal matrix $P$, a (rotational) change of basis, that brings $Q$ into block diagonal form, see details here.