Find eigenvalues and eigenvectors of the operator $A$ The question is: Find the eigenvalues and eigenvectors of the operator $A$ on $\Bbb{R}^3$ given by $A\mathbf{x}=|\mathbf{a}|^2 \mathbf{x}- (\mathbf{a} \cdot \mathbf{x}) \mathbf{a}$, where $\mathbf{a}$ is a given constant vector. How do you know without any calculations that $A$ must have an orthonormal eigenbasis? 
I have seem examples similar to this question. I'm wondering if there's any systematic way to solve this kind of questions. Someone showed to me that you first get $\mathbf{x}=\lambda\mathbf{a}$. What is the reason behind that and how does this help to solve the problem? 

Note that for all $x,y\in\mathbb{R}^3,$ $$\langle A x,y\rangle=\langle|a|^2x-\langle a, x\rangle a,y\rangle=|a|^2\langle x,y\rangle-\langle a,x\rangle\langle a,y\rangle=\langle x,Ay\rangle,$$in other words $A$ is self adjoint, hence it has an orthonormal eigenbasis.
In fact, a simple calculation shows that $a$ is an eigenvector, and so is every vector perpendicular to $a$.
