We define a linear transformation $\displaystyle T:Mat_{m,m}->Mat_{m,m}$ of $\displaystyle m\times m$ matrices such that $\displaystyle T(M)=M^{Tr}$.

Basically we are looking at the transposition of a matrix, treating it as a linear transformation. I need to find characteristic polynomials, eigenvalues, eigenvectors, etc. I started by looking at 2-by-2 and 3-by-3 to get some insight. I attached the calculations.

The 2x2 and 3x3 helped a little, but I still cannot figure out how to generalize it.