Understanding a proof: Eigenvalues of a real symmetric matrix are real I'm trying to understand the following proof, but I have two questions about it that I hope someone could clarify to me: 
1) How does the last equation ($\lambda ^t \overline{Z} Z = \overline{\lambda}^t Z \overline{Z}$) follow from the previous two equalities?
2) Where did they use that $A$ is a real symmetric matrix?
Thanks for your help.

 A: It's not difficult to do it in the more general case of a hermitian matrix; just change $H$ (for the conjugate transpose) into $T$ for the transpose, if you don't trust complex numbers.
The only things to remember are that, for arbitrary matrices and scalars, we have
$$
(AB)^H=B^HA^H,\quad
(\alpha A)^H=\bar{\alpha}A^H
$$
(where $\bar{\alpha}$ is the conjugate of the complex number $\alpha$).
So, let $A=A^H$ be a hermitian matrix; let $v$ be an eigenvector (column vector) relative to the (complex) eigenvalue $\lambda$: $Av=\lambda v$. We have
$$
\lambda(v^Hv)=v^H(\lambda v)=v^HAv=v^HA^Hv=(Av)^Hv=(\lambda v)^Hv=\bar{\lambda}(v^Hv)
$$
Since $v^Hv\ne0$, because $v\ne0$, we get
$$
\lambda=\bar{\lambda}
$$
so $\lambda$ is real.
If you don't want to start from $\lambda(v^Hv)$, just observe that
$$
v^HAv=v^HA^Hv
$$
and develop each side using the fact that $Av=\lambda v$.
A: Let's begin with this equation
$$Z^tA^t\overline{Z} = \overline{Z}^tAZ = \lambda\overline{Z}^tZ$$
The desired outcome of this equation is the equality of the first and last entries:
$$Z^tA^t\overline{Z} = \lambda\overline{Z}^tZ$$
At this point we take the conjugate of this equation to obtain
$$\overline{Z}^t\overline{A}^tZ = \overline{\lambda}Z^t\overline{Z}$$
Now we use the assumption that $A$ is real and symmetric to conclude that $\overline{A}^t = A$. Whence we get
$$\overline{Z}^tAZ = \overline{\lambda}Z^t\overline{Z}$$
In the first term, we have $AZ = \lambda Z$, hence
$$\lambda\overline{Z}^tZ = \overline{\lambda}Z^t\overline{Z}$$
In the proof you showed, the steps of taking the conjugate and using the symmetry of the matrix were implicit.
