Is matrix $B$ an image of matrix $A?$

Question

Consider the following two matrices:

$$A=\begin{pmatrix} e^s&0\\ 0&e^{-s} \end{pmatrix}$$

$$ B=\begin{pmatrix} e^{e^s} &0\\ 0&e^{e^{-s}} \end{pmatrix} $$

Is $B$ an image of $A?$ If we let $e^s\leftrightarrow s$ then does this imply that $B$ is an image of $A?$ I'm a little confused about this.

Answer 1

Since $B=e^A$, the matrix $B$ is the image of $A$ by the exponential map.