eigenvalues of sum of a matrix and its conjugate transpose We know that the eigenvalues of a matrix and its conjugate transpose are conjugate to each other. My question is: If $\lambda$ is an eigenvalues of $A$, what is the eigenvalue of $A+A^*$ in terms of $\lambda$, where $A^*$ denotes the complex conjugate of $A$

Your question can't possibly work / be meaningful.  
Consider $A$ as the strictly upper triangular matrix with all ones above the diagonal.  Then $A + A^*$ is real symmetric and hence diagonalizable (it in fact has eigenvalues of $n-1$ and $-1$, though that's outside the scope).  Yet all eigenvalues of $A$ were zero.  You can create your own examples by selecting any upper triangular matrix $A$ and then looking at the eigenvalues of $A + A^*$.  
There's no reason to think that eigenvalues directly add or have some other simple relationship for non-commuting matrices.  
