Let $A$ be a diagonalizable matrix, i.e., $A=SDS^{−1}$. Recall that columns of $S$ correspond to eigenvectors of $A$, and the diagonal entries of the diagonal matrix $D$ correspond to its eigenvalues.

Is it possible to choose $S$ such that the condition number of $S$ is bounded, i.e., can we choose $S$ such that $|S| |S^{−1}|$ is bounded?