Diagonal Transformation

Question

Given an orthogonal matrix $r \in SO(n,\Bbb R)$), and some diagonal matrix $d$ of the same size $n \times n$;
Is the matrix $r\cdot d\cdot r^{-1}$ also diagonal?

Answer 1

No. $r$ denotes a change of basis which is orthogonal. And there is no reason for $d$ to remains diagonal.

Example $d=\begin{pmatrix}1&0\\0&2\end{pmatrix}$ and $r$ a rotation of angle $\pi/4$.

Answer 2

As was answered by user126154, the answer is no. More generally, one of the main theorems about complex matrices states that a complex matrix $A$ is diagonalizable by a unitay matrix iff $A$ is normal, namely $AA^* =A^*A$. For matrices over the real numbers, you need the eigenvalues to be real, and this happens iff $A=A^*$, and since $A$ is a real matrix we have that $A=A^T$. In particular you get that any symmetric matrix is diagonalizable by an orthogonal matrix.