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? 

With matrix multiplication notation you get
$$Ax = (a^Ta)x-(aa^T)x = ((a^Ta)I-aa^T)x$$
so
$$A = (a^Ta)I-aa^T$$
where you can see that $A$ is symetric and therefore has orthogonal eigenvectors (look up the property on wikipedia).
Eigenvalues $\lambda$ fulfill
$$Ax = \lambda x$$
or in your case
$$((a^Ta - \lambda)I-aa^T)x = 0 \ .$$
You can find the eigenvalues by computing the roots of the polynomial
$$\text{det}((a^Ta - \lambda)I-aa^T) = 0 \ .$$
