Wednesday, April 30, 2014

Rotated matrix and rotated eigenvectors.

Rotated matrix and rotated eigenvectors.

Let $R_\theta$ be a rotational matrix, my intuition (from experiences with quadratic forms) tells me that if $A$ has an eigenvector $v$, then $R_\theta^T A R_\theta$ has an eigenvectors $R_\theta^T v$.

Is this true? If so, how can I prove this?

The matrix $R_\theta^TAR_\theta$ has eigenvector $R^T_\theta v$, because $$R_\theta^TAR_\theta(R_\theta^Tv)=R^T_\theta A(Iv)=R^T_\theta Av=R^T_\theta(\lambda v)=\lambda (R^T_\theta v)$$ where I've used the fact that $R_\theta^TR_\theta=I$, because $R_\theta$ is an orthogonal matrix.

Addtionally, this also means if any $\lambda$ is an eigenvalue of A then it's also an eigenvalue of $R_\theta^T A R_\theta$, i.e. all eigenvalues are preserved.

Actually, this is not only true for rotational matrices, as long as $R^TR = I$, (R is orthonormal), this will work!