If a matrix representation of a linear transformation is similar to a diagonal matrix, why does this imply that the Jordan normal form must also be diagonal?
[Math] Why does similarity with a diagonal matrix imply that the Jordan normal form must also be diagonal
diagonalizationjordan-normal-formlinear algebralinear-transformationsmatrices
Best Answer
The comment by Paul Sinclair gives the fundamental reason why this is so: the Jordan normal form is a form that generalises the diagonal form such that all complex matrices are similar to some Jordan normal form, which is unique up to permutation of the blocks; diagonalisable matrices already were similar to some diagonal matrix, so that must be their Jordan normal form.
But explicitly, let $A$ have (i.e., be similar to) a Jordan normal form $J$ that has at least one Jordan block of size larger than$~1$ (so that it is not diagonal). By permutation of the blocks we may assume the first Jordan block has size${}>1$, and say it has $\lambda$ as diagonal entries; then the second standard basis vector $e_2$ satisfies $(J-\lambda I)e_2=e_1\neq0$ but $(J-\lambda I)^2e_2=0$. Since $J$ is similar to $A$, the vector $v$ corresponding to$~e_2$ under the change of basis satisfies $(A-\lambda I)v\neq0$ but $(A-\lambda I)^2v=0$ which shows that $A$ is not diagonalisable (since for diagonalisable $A$ one has that $\ker(A-\lambda I)^2$ is equal to the eigenspace $\ker(A-\lambda I)$ for$~\lambda$).