Sum of unitary transformation I am having struggle with this question.
suppose I have two unitary matrices.
Is their sum is  normal ?
I am try to give an example to show it is not true and I can not find.
I try to proof and I reach this results:
$T$,$S$ are unitary then:
$(T+S)(T+S)^{*}=(T+S)^{*}(T+S)$
$(T+S)(T^{*}+S^{*})=(T^{*}+S^{*})(T+S)$
$(T+S)T^{*}+(T+S)S^{*}=(T^{*}+S^{*})T+(T^{*}+S^{*})S$
$TT^{*}+ST^{*}+TS^{*}+SS^{*}=T^{*}T+S^{*}T+T^{*}S+S^{*}S$
$ST^{*}+TS^{*}=S^{*}T+T^{*}S$
and I can not continue from here.
I can not tell is this equation is true or not.
and BTW 
matrix is unitary iff matrix is symmetric ?
Thanks in advanced !!
 A: First, symmetric matrix and unitary matrix are two unrelated concepts. It is easy to find symmetric matrices that are not unitary, and vice versa. 
For an example where $T+S$ is not normal, we can take $T=\begin{bmatrix} 0 & i \\ -i & 0\end{bmatrix}$, and $S=\begin{bmatrix} \lambda_1 & 0 \\ 0 & \lambda_2\end{bmatrix}$ with $\lambda_i\bar{\lambda}_i=1$ for $i=1,2$. The calculations are omitted.
It is interesting to note however, that if both $S$ and $T$ are taken in $SU(2)$ (i.e., $2\times 2$ unitary matrices whose determinant is one), then indeed $S+T$ is normal.  One way to see this is to identify $SU(2)$ with the norm 1 elements in the Hamilton quaternion algebra ( then the *-operator corresponds to taking conjugate).  Then $$ST^*+TS^*=ST^*+(ST^*)^*=\mathrm{Tr}(ST^*),$$ and similarly $$T^*S+S^*T=\mathrm{Tr}(T^*S).$$
Now $\mathrm{Tr}(ST^*)=\mathrm{Tr}(T^*S)$. 
Hmm, I guess I am only using the fact that $X+X^{-1}=\mathrm{Tr}(X)$ for any $X\in SL_2(\mathbb{C})$ using the formula $\begin{bmatrix} a & b \\ c & d\end{bmatrix}^{-1}=\frac{1}{ad-bc}\begin{bmatrix} d & -b \\ -c & a\end{bmatrix}$. 
