# Eigenvalues of symmetric complex matrices .

Is it true that for symmetric matrix with complex entries all eigenvalues are real.

I have seen the proof for Hermitian matrices and proved it for real symmetric matrices,but for complex symmetric matrices I dont know how to determine this.

Thank you in advance for your help.

• Hint: Try a 2x2 matrix where all entries = $i$. – Amzoti Mar 5 '13 at 5:09
• This is true for self-adjoint matrices, aka hermitian matrices. Not for symmetric matrices. Look at $\left(\matrix{0&i\\i&0}\right)$ for instance. – Julien Mar 5 '13 at 5:10
• @julien: easier example than mine! Regards – Amzoti Mar 5 '13 at 5:12
• How about $\begin{bmatrix} i \end{bmatrix}$. No simpler than that! – copper.hat Mar 5 '13 at 5:54
• @copper.hat: Nice example! – Amzoti Mar 5 '13 at 6:06