Eigenvalue spectrum of a diagonal matrix times a symmetric matrix

Question

Consider a symmetric matrix $A$ with eigenvalues $\lambda_{1}$, $\lambda_{2}$, $\lambda_{3}$, $\lambda_{4}$, $\lambda_{5}$, $\cdots$, $\lambda_{n}$. $n$ being the dimension of the matrix.

Suppose I have another matrix, $B$ with diagonal entries, $0$, $\exp(\text{i} \theta)$, $\exp(\text{i} 2 \theta)$, $\exp(\text{i} 3 \theta)$, $\exp(\text{i} 4 \theta)$, $\exp(\text{i} 5 \theta)$, $\cdots$, $\exp(\text{i} (n - 1) \theta)$. Rest all terms are zero in the matrix.

Now I multiply, $C = A \cdot B$. Can I comment on the eigenvalues of $C$?

Answer 1

Other than the potential typo that the comment above mentioned, this post Eigenvalues of product of a matrix with a diagonal matrix seems to be the same problem you have.