Let $T : M_{n \times n}(\Bbb R) \to M_{n \times n}(\Bbb R)$ be the function given by $T(A)=A^t$ (the transpose of $A$).
I need to find the minimal polynomial and the characteristic polynomial of $T$. So, to find the characteristic polynomial, I'm trying to find the associated matrix of $T$. I did it for the case $n=2$. The coordinates of a matrix $\left( \begin{array}{ccc}
a & b \\
c & d \end{array} \right)$ in the canonical basis $\beta$ is $[X]_\beta=\left( \begin{array}{c}
a \\
b \\
c \\
d \end{array} \right)$. Let
$$A = \left( \begin{array}{ccc}
1 & 0 & 0 & 0 \\
0 & 0 & 1 &0 \\
0 & 1 & 0 & 0 \\
0 & 0 &0 & 1 \end{array} \right)$$
Then
$$A[X]_\beta=[A^t]_\beta==\left( \begin{array}{c}
a \\
c \\
b \\
d \end{array} \right).$$
So, $A$ is the associated matrix of $T$. I know I can do the same for any $n$, but I don't know how to generalize it, and I need it to find the characteristic polynomial, solving $\det (\lambda I-A)=0$. Maybe there's an easier way to find the characteristic polynomial, if you know it, please let me know. If not, how can I generaize this matrix $A$ for any $n$ to find $\det (\lambda I-A)=0$? Thanks so much for your help,
Best Answer
Simply note that $T^2=I$ but $T\ne I$.
This gives you the minimal polynomial of $T$. The characteristic polynomial has the same irreducible factors as the minimal polynomial and degree equal to the dimension of the space $T$ acts on.