[Math] Find an ordered basis $B$ for $M_{n\times n}(\mathbb{R})$ such that $[T]B$ is a diagonal matrix for $n > 2$

Question

I have a homework problem that I'm stuck on. It is problem 5.1.17 in the Friedberg, Insel, and Spence Linear Algebra book for reference.

"Let T be the linear operator on $M_{n\times n}(\mathbb{R})$ defined by $T(A) = A^t$" is the beginning of the problem.

The part I'm concerned with is part d, "Find an ordered basis $B$ for $M_{n\times n}(\mathbb{R})$ such that $[T]B$ is a diagonal matrix for $n > 2$?"

I completed part c), which is the same part but for $2\times2$ matrices instead.

I'm stuck on figuring this part out.

Answer 1

Hint: What matrices satisfy $A^\top =\lambda A$ for some $\lambda$?