# Given n-dimensional orthonormal basis with positive orientation, can I reach the standard basis via a sequence of rotations?

Given a n-dimensional orthonormal basis in Euclidean space with positive orientation, can I reach the standard basis via a sequence of rotations?

• sequence of rotations? Commented Jun 2, 2017 at 4:48
• Some define a (proper) rotation matrix as an orthogonal matrix with positive determinant. With that definition of rotation it only takes one to reach the standard basis.
– amd
Commented Jun 2, 2017 at 6:00

Yes. The matrix going between these two bases is in the special orthogonal group and is conjugate there to a matrix with block decomposition $$\pmatrix{R_{t_1}&&&\\&R_{t_2}&&\\&&\ddots&\\&&&R_{t_m}}$$ or $$\pmatrix{R_{t_1}&&&&\\&R_{t_2}&&&\\&&\ddots&&\\&&&R_{t_m}\\&&&&1}$$ where $R_t$ is a rotation matrix in two dimensions.
• @JohnZheng It's eigenvalues and eigenvectors. They are complex in general but if you take the real and imaginary parts of a complex eigenvalue you get a 2-dimensional subspace where the matrix acts as a 2-dimensional rotation. You get an orthogonal decomposition into such subspaces (unless $n$ is odd where you get one-dimension left over). Commented Jun 2, 2017 at 5:23