# Eigenvalues of a $3\times 3$ symmetric matrix [duplicate]

Given a $3\times 3$ symmetric matrix

$$M= \begin{pmatrix} A & B & C \\ B & D & E \\ C & E & F\\ \end{pmatrix},$$

how do I find the eigenvalues? Brute-forcing it by looking at the roots of the characteristic polynomial seems overwhelming.

Are there any properties of symmetric matrices that might simplify the problem or am I overlooking something?

## marked as duplicate by Jennifer, Watson, egreg, Kushal Bhuyan, mickepJun 13 '16 at 12:34

• @Lordofdark very relevant, but it doesn't provide a definitive answer to the question. – Omnomnomnom Jun 13 '16 at 12:21
• Hi. Yes, I read it before I posted, but like @Omnomnomnom stated, they do not provide a clear answer. – Henrymerrild Jun 13 '16 at 12:22

In the general$^*$ case there is no escape, you must solve a cubic equation. (I am not talking about particular cases such that the coefficients and roots are rational or solutions can be spotted by inspection.)
$^*$Note that the matrix has $6$ independent coefficients. This is more than enough to correspond to a general cubic polynomial.