What will be the eigenvalues of the antisymmetric matrix $ A=\begin{pmatrix} 0 & -n_3 & n_2 \\n_3 & 0 & -n_1 \\-n_2 & n_1 & 0\end{pmatrix} $, where $ n_1$, $n_2$ & $n_3 $ are the components of a unit vector?
I need the way to solve it.
What will be the eigenvalues of the antisymmetric matrix $ A=\begin{pmatrix} 0 & -n_3 & n_2 \\n_3 & 0 & -n_1 \\-n_2 & n_1 & 0\end{pmatrix} $, where $ n_1$, $n_2$ & $n_3 $ are the components of a unit vector?
I need the way to solve it.
Hint: Your matrix being a antisymmetric of odd order, should have $0$ as an eigenvlaue (why?). Now from the trace condition, you see that the remaining two have opposite sign. So, you need to calculate only the coefficient of $\lambda$ in the characteristic equation, which is sum of the three $2\times2$ principle minor. If you calculate it and use your condition $|\vec{n}|^2=1$, it will be a very well known number....