Let $ N=\begin{pmatrix}0&1&&\\&\ddots&\ddots&\\&&0&1\\&&&0 \end{pmatrix}_{n\times n} $ and $I$ is the identity matrix of order $n$. How to prove $I+N\sim e^N$?
Clarification: this is the definition of similarity, which is not the same as equivalence.
Update:
I noticed a stronger relation, that $A\sim N$, if $ A=\begin{pmatrix}0&1&*&*\\&\ddots&\ddots&*\\&&0&1\\&&&0 \end{pmatrix}_{n\times n} $ and $*$'s are arbitrary numbers.