Similarity of linear transformations

Question

Suppose there are two linear transformations $A$ and $B$ on the same finite dimensional vector space $V$, such that $\dim Im(A) = \dim Im(B)$. Is it always true that they are similar. What about the converse?

Answer 1

1. Similar linear transformations must have the same eigenvalues, with the same multiplicities.

2. Linear transformations whose images have the same dimension must have the same rank, i.e. the same number of nonzero eigenvalues.

Hence $1\to 2$ but $2\not \to 1$.

Answer 2

No. Notice that the identity matrix $I_n$ is similar only to itself so take for a counterexample $A=I_n$ and $B$ any invertible matrix different to $I_n$. The converse is true because if $P$ is invertible then:

$$\operatorname{rank}PA=\operatorname{rank}A$$

Answer 3

consider the diagonal matrix $D = \left( \begin{array}{ll} 1 & 0 \cr 0 & 0\end{array} \right) $ and the nilpotent matrix $N = \left( \begin{array}{ll} 0 & 1 \cr 0 & 0 \end{array} \right).$ both have the same column space(image) but they are not similar because $D$ has eigenvalues $0$ and $1$ but $N$ has eigenvalues $0$ and $0.$ (similar matrices have the same eigenvalues)