I'm working on the following problem:
Let A be a reflection matrix, such that, $a_{ij}=\delta_{ij}-2n_{i}n_{j}$, about a plane perpendicular to $\vec{n}$, $\vec{n}$ being the unitary vector. Find its eigenvalues and eigenvectors algebraically.
My first thought was on using $A\vec{v}=\lambda\vec{v} \Rightarrow (A-\lambda I)\vec{v}=0$, but I got an absolute huge equation, given that the matrix was
\begin{bmatrix}
1-2n_1^2-\lambda & -2n_1n_2 & -2n_1n_3 \\
-2n_2n_1 & 1-2n_2^2-\lambda & n_2n_3 \\
-2n_3n_1 & n_3n_2 & 1-2n_3^2-\lambda \\
\end{bmatrix}
So, $det(A)$ has given a lot of terms to manipulate. I was thinking if there is a different approach to this problem, maybe using determinant properties or writing the matrix on a different basis, but could not develop any further.
Any tips?
Best Answer
Hint: think geometrically.
What does $A$ map $\vec n$ itself to ?
A vector $\vec m$ such that $\vec m . \vec n = 0$ is the in the plane of the reflection. So what does $A$ map $\vec m$ to ?