Problems in Linear Algebra There is this rather hard linear algebra problem in my notes that the professor left as an exercise during the holidays if we have free time, and it goes:

Problem: Let $M_n(\mathbb{R})$ be the vector space of all $n \times n$ matrices over the real number line. For $A \in$ $M_n(\mathbb{R}),$ let $A^t$ denote the transpose of matrix $A$. Define the linear operators $\tau_A: M_n(\mathbb{R}) \rightarrow  M_n(\mathbb{R})$ by $\tau_A(X) = AXA^t$. Prove that $\text{Tr}(\tau_A)$ is $\text{Tr}(A)^2$ and $\text{Det}(\tau_A)$ is $\text{Det}(A)^{2n}.$

Can anyone give me some start so I can get going with this problem? I would appreciate the help. 
 A: There are two basic approaches to this problem.  One is to find the eigenvalues/eigenvectors of the transformation, and then to compute the trace/determinant using the product/sum (I think this is what Qiaochu has in mind).  The other is to try to interpret the trace/determinant in the context of maps on $M_n(\Bbb R)$.
Hint: (For approach 1) Suppose that $A$ is diagonal.  What do the eigenvalues/eigenvectors of $A$ look like?  Consider in particular the case where all eigenvalues of $A$ are distinct.  Now, what do the eigenvectors of $A$ look like if $A$ is diagonalizable?
Once you have the result for diagonalizable matrices, you can get the general result either using the continuity of trace/determinant, or by extending our argument above using Jordan form.
Hint: (For approach 2) We may compute the trace of a map $\tau:M_n(\Bbb R)\to M_n(\Bbb R)$ as
$$
\sum_{i,j = 1}^n \operatorname{trace}(E_{ij}^T \tau(E_{ij}))
$$
where $E_{ij}$ denotes the matrix with a $1$ in the $i,j$ entry and zeros elsewhere.
To compute the determinant, it is helpful to consider this map as a composition.  Namely, $\tau_A = \tau^{(1)}_A \circ \tau^{(2)}_A$, where
$$
\tau^{(1)}(X) = AX, \qquad \tau^{(2)}(X) = XA^T
$$
If you find the matrices of $\tau^{(1)}$ and $\tau^{(2)}$ relative to the basis $\{E_{ij}: 1 \leq i,j \leq n\}$ (taken in lexicographical order), then you will find that it is easy to compute $\det \tau^{(1)}$ and $\det \tau^{(2)}$ by exploiting the block-structure of these matrices.
