Is there a way to pick out the transpose $\lambda M \mathop. M^T$ (with $\lambda$ denoting an anonymous function) from the automorphisms of linear transformations from $\mathbb{R}^n$ to $\mathbb{R}^n $, or, equivalently, linear transformations of $\mathbb{R}^{n \times n}$ ?
I guess what I'm asking is: is there a way to pick out the transpose using either a) the structure of the group of automorphisms or b) properties like "being an involution" or "preserving the characteristic polynomial" that are independent of choice of basis and do not explicitly refer to the cells of the matrix?
I remember part of a remark from a professor who said that the complex conjugate $\lambda z \mathop. \overline{z}$ and the identity map $\lambda z \mathop. z$ are the only field automorphisms of $\mathbb{C}$, with some additional language to rule out wild automorphisms of $\mathbb{C}$ . I forget the exact language used to rule out wild automorphisms. I think it works if you restrict your attention to automorphisms that fix $\mathbb{R}$, but that seems a bit circular (conceptually circular, not a circular definition).
Anyway, once the wild automorphisms are ruled out, then $\lambda z \mathop. \overline{z}$ is the only nontrivial automorphism of $\mathbb{C}$.
Since a complex number $a + b\iota$ can be represented as the $2\times 2$ real matrix below, it seems plausible that the transpose might be distinguishable from the outside for the more general case of square real matrices.
$$ \begin{bmatrix} a & b \\ -b & a \end{bmatrix} $$
