proof on similarity of matrices Could you please help me with the following problem?
Let $A$ be an $n$$\times$$n$ complex matrix. Prove that $A$ is similar to $B$, which is an $n$ $\times$ $n$ real matrix, if and only if $A$ is similar to its conjugate transpose.
I have been trying to solve it for quite a long time, I tried to expand the definition of similarity but I did not  manage to get anything useful. I have no better idea.
 A: The following facts are useful:


*

*(1) For any $A\in\mathbb{C}^{n\times n}$, $A\sim A^T$.

*(2) For any $A\in\mathbb{C}^{n\times n}$, $A\sim B$ for some $B\in\mathbb{R}^{n\times n}$ if and only if $A\sim\bar{A}$.


So if $A\sim B$ for some real $B$, then 
$$A\stackrel{(2)}\sim\bar{A}\stackrel{(1)}\sim\bar{A}^T=A^*.$$
On the other hand, if $A\sim A^*$, then
$$
A\sim A^*=\bar{A}^T\stackrel{(1)}\sim\bar{A}\stackrel{(2)}\sim B
$$
for some real $B$.

Fact (1) can be shown by establishing a similarity transformations between elementary Jordan blocks and their transposes. If $J_{\lambda}$ is an elementary Jordan block and $R$ is the "reversal" matrix (an identity with flipped columns (or rows)), then $R^{-1}J_{\lambda}R=J_{\lambda}^T$ (note that $R^{-1}=R^T$).
For fact (2), the tricky part is to show that if $A\sim\bar{A}$, then $A$ is similar to a real matrix (the other direction of the equivalence is easy). If $A\sim\bar{A}$, then both $A$ and $\bar{A}$ have the same Jordan form $J$. But since both $A$ and $\bar{A}$ are similar to $J$, for each elementary Jordan block $J_{\lambda}\in\mathbb{C}^{k\times k}$, there must be a "matching" Jordan block $J_{\bar{\lambda}}$. The tricky part is to show that the "paired" Jordan blocks
$$
\pmatrix{J_{\lambda}&0\\0&J_{\bar{\lambda}}}
$$
are similar to a real matrix of the form
$$
\pmatrix{K&I_2&&\\
          &K&I_2&\\
          &&\ddots&\ddots\\
          &&&K&I_2\\&&&&K}\in\mathbb{R}^{2k\times 2k},
\quad
K:=\pmatrix{\Re\lambda&\Im\lambda\\-\Im\lambda&\Re\lambda},
$$
where $I_2$ is the $2\times 2$ identity (this leads to the so-called real Jordan form).
