Name for matrix operation that inverses off-diagonal elements

Question

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}) $

Answer 1

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 $(b,c)\neq (0,0)$.

Answer 2

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).