Entries of a symmetric matrix

Question

What is difference between a symmetric matrix over C and hermitian matrix. Or we have to take entries from R only to define symmetric matrix. What I am understanding is every real symmetric matrix is hermitian but what about complex symmetric matrix?
If I don't use conjugate then complex symmetric matrix need not to be hermitian at all. Am I right?

Answer 1

Yes you are correct.

$$ \begin{pmatrix} 2*i&1+i\\ 1+i&0 \end{pmatrix} $$

is symmetric. But because of the $2*i$ not being real, and the $1+i$ instead of a $1-i$ it is not Hermitian.

Answer 2

The difference is that a Hermitian matrix is symmetric with respect to the complex inner product. That is, $$ \langle Ax,y\rangle = \langle x,A^*y\rangle = \langle x,Ay\rangle, $$ where $A^*$ is the conjugate transpose of $A$ and where $x,y$ are vectors in $\mathbb{C}^n$. Such a matrix has only real eigenvalues because, if $Ax=\lambda x$, then $$ (\lambda-\overline{\lambda})\|x\|^2=\langle \lambda x,x\rangle-\langle x,\lambda x\rangle=\langle Ax,x\rangle-\langle x,Ax\rangle = 0.$$