# linear algebra – Condition number for matrix of eigenvectors of a diagonalizable matrix

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?