General formula for eigenvectors of a 3x3 matrix Sorry if this is a dumb question but given a general 3x3 matrix
$A = \begin{pmatrix} a & b & c \\
d & e & f \\
g & h & i
\end{pmatrix} $
and assuming it has 3 distinct eigenvalues $\lambda_1, \lambda_2, \lambda_3$, is there a general (analytical) formula for the eigenvectors of this matrix?

I assume the underlying field is $\mathbb{R}$.
In the case you describe - distinct eigenvalues - the vector product
$$
v_{\lambda}:=(d, e-\lambda, f)\wedge (g,h,i-\lambda)
$$
is a $\lambda$-eigenvector for $\lambda=\lambda_1,\lambda_2,\lambda_3$.
This is because it is perpendicular to $(d, e-\lambda, f)$, and $(g,h,i-\lambda)$; and also to $(a-\lambda, b,c)$ - which is a linear combination of $(d, e-\lambda, f)$, and $(g,h,-\lambda)$ in the case when $\lambda$ is an eigenvalue.
