Name for matrix operation that inverses off-diagonal elements Is there any operation that sends matrix
$$
S=\left(\begin{array}{cc} a&b\\
c& d
\end{array}\right) \to S' 
=\left(\begin{array}{cc} \:\:\:a&-b\\
-c& \:\:\:d
\end{array}\right)
\quad a,b,c,d \in \mathbb{R}
$$
by setting off-diagonal elements of 2x2 matrix to inverse?
My question is: is there a name (or notation) for this kind of matrix operation?
If I would assume that $b,c \in Im$ (imaginary) then the operation obviously can be defined as conjugate $S'=(S^*)^T$. But all values are real in the given case.
Edit: $S \in SL(2, \mathbb{R}) $
 A: The two matrices $S$ and $S'$ are conjugate, or similar since they have the same characteristic polynomial
$$
t^2-(a+d)t+(ad-bc),
$$
provided that they aren't a scalar multiple of the identity, i.e., provided that $(b,c)\neq (0,0)$.
A: Let me post my own answer.
Let $S \in \mathrm{SL(2,\mathbb{R}})$, and let C be a group of matrices with two imaginary off-diagonal elements- a subgroup of $\mathrm{SL(2,\mathbb{C}})$.
Since the S can be isomorphically mapped to some element $A \in C$ using the map
$S \to \mathrm{A=T S T^{-1}}$ as following
$$
S \to A=\left(\begin{array}{cc} \sqrt{i}&0\\ 0& \sqrt{-i} \end{array}\right)  \left(\begin{array}{cc} a&b\\ c& d \end{array}\right)  \left(\begin{array}{cc} \sqrt{-i}&0\\ 0& \sqrt{i} \end{array}\right) = \left(\begin{array}{cc} \:\:\:a&ib\\ -ic& d \end{array}\right)
$$
(it is easy to see that the map is bijective, multiplicative e t c and the isomorphism).
Then after conjugating $A \to A^*$, the inverse map $\mathrm{T^{-1} (A^*) T }$ sends $A^*$ to
$$
S'=\left(\begin{array}{cc} \sqrt{-i}&0\\ 0& \sqrt{i} \end{array}\right)  \left(\begin{array}{cc} a& -ib\\ ic& \:\:\:d \end{array}\right)  \left(\begin{array}{cc} \sqrt{i}&0\\ 0& \sqrt{-i} \end{array}\right) = \left(\begin{array}{cc} \:\:\:a&-b\\ -c& \:\:\:d \end{array}\right)
$$
The operation is thereby is complex conjugation in the isomorphic group C (a subgroup of $\mathrm{SL(2,\mathbb{C}}$) with two imaginary off-diagonal elements). And since it is conjugation in the isomorphic group it is the conjugation.
PS. The same is valid for the conjugate transpose (or Hermitian transpose).
