Let $A$ be non-zero matrix such that $a_{ij}=0$ $\forall i\ge j$. $D$ be a diagonal matrix with distinct diagonal entries. Now I want to show that $D$ is similar to $D+A$.
Then how can I show that this does not hold without "Distinct diagonal entries" assumption?
My try:
I am thinking in terms of "change of basis". So $D$ represents a linear transformation which has distinct eigen vectors with distinct eigen values. Now $A$ is also nilpotent. I can't assemble these facts to get through the problem.
Can anyone help or suggest me anything? Thanks in advance.