What is $e^{A}$ where A is an anti-diagonal matrix I am trying to get a closed form for the matrix produced by the following operation: $$e^A$$ where $A$ is an anti diagonal matrix, say, of size $2\times 2$:
$$A=\begin{pmatrix}
0 &b \\ 
c &0 
\end{pmatrix}$$
Using Mathematica MatrixExp I got
$$ \left(
\begin{array}{cc}
 \frac{1}{2} e^{-\sqrt{b} \sqrt{c}}+\frac{1}{2} e^{\sqrt{b} \sqrt{c}} & \frac{\sqrt{b} e^{\sqrt{b} \sqrt{c}}}{2 \sqrt{c}}-\frac{\sqrt{b} e^{-\sqrt{b} \sqrt{c}}}{2 \sqrt{c}} \\
 \frac{\sqrt{c} e^{\sqrt{b} \sqrt{c}}}{2 \sqrt{b}}-\frac{\sqrt{c} e^{-\sqrt{b} \sqrt{c}}}{2 \sqrt{b}} & \frac{1}{2} e^{-\sqrt{b} \sqrt{c}}+\frac{1}{2} e^{\sqrt{b} \sqrt{c}} \\
\end{array}
\right)
$$
But here one can find a formula for the computation on a general $2\times 2$ matrix, at the bottom of the page. Using that formula I got a different result.
What is the correct answer?

Recall that: $$e^A=\sum_{k=0}^{+\infty}\frac{A^k}{k!}.$$
Moreover, notice that in your case, one has: $$A^2=\begin{pmatrix}bc&0\\0&bc\end{pmatrix}.$$
Therefore, you can compute $e^A$ summing on even and odd integers. Indeed, one has: $$A^{2k}=\begin{pmatrix}b^kc^k&0\\0&b^kc^k\end{pmatrix},A^{2k+1}=\begin{pmatrix}0&b^{k+1}c^k\\b^kc^{k+1}&0\end{pmatrix}.$$
