We consider the eigenvalue problem for octonionic $3\times3$ Hermitian matrices (the exceptional Jordan algebra, also known as the Albert algebra). For real eigenvalues, most of the properties expected by analogy with the complex case still hold, provided they are reinterpreted to take into account of the lack of commutativity and associativity. There are nevertheless some interesting surprises along the way, among them the existence of nonreal eigenvalues, and the fact that the components of primitive idempotents (elements of $OP^2$, the Cayley--Moufang plane) always associate. Applications of these results will be briefly discussed, both to the study of exceptional Lie groups (the Albert algebra is the minimal representation of $e_6$) and to physics ($OP^2$ can be interpreted as the solution space of the Dirac equation in 10 spacetime dimensions).