General form for Eigenvector of a 3 by 3 symmetric matrix

Question

I am looking for a general form for both Eigenvalues and Eigenvectors for an arbitrary 3 by 3 symmetric matrix. Is there a compact form present in the literature? I have tried using Mathematica and it seems pretty messy even after simplification.

Answer 1

The eigenvalues are the roots of the characteristic polynomial, which is a cubic whose coefficients are polynomials in the entries of the matrix. There are formulas for the roots of a cubic, but they are messy. It's not going to get less messy if you substitute in the expressions for the coefficients in terms of the matrix entries. The eigenvectors form a basis of the null space of $A - \lambda I$ where $A$ is your matrix and $\lambda$ an eigenvalue. Still more messy.

Here's one help for the last step, though. Assume there are three distinct eigenvalues $\lambda_1$, $\lambda_2$, $\lambda_3$. An eigenvector for eigenvalue $\lambda_1$ is $(A - \lambda_2 I)(A - \lambda_3 I) x$ where $x$ is any vector that is not an eigenvector for $\lambda_2$ or $\lambda_3$.