# Eigen-properties of a Rotation Matrix

Let R be a rotation matrix and . Usually we don’t talk about rotation matrix in higher space, though the definition of the rotation matrix is not constrained on 2D or 3D.

**1. Eigenvalues and Eigenvectors:**

Consider the rotation axis of R is x, then Rx=x. So R must have an eigenvalue as 1. The corresponding eigenvector is the rotation axis.

The other eigenvalues are all complex. And . That means the eigenvalues is the root of . For 3D, the eigenvalues are

**2. Singular values and Singular vectors:**

An SVD of A is . The singular values of A is the square root of the eigenvalues of and . If A is a rotation matrix, then . So the eigenvalues of is 1. So the singular values of is all 1. So the condition number of a rotation matrix is

So the numerical stability of R is the best!

The singular vectors of A is respectively the eigenvectors of and .

**Questions remained:**

(1) why does the eigenvalues of R be the root of x^d=1?

(2) the eigenvector of R associating its complex eigenvalue is real or complex?