Spectral analysis of a $2\times 2$ matrix

Question

We have the matrix

$$ A=\begin{pmatrix}-\sqrt{3} & 3 \\ 3 &\sqrt{3}\end{pmatrix}. $$

I want to find the spectral analysis of that matrix, i.e., write $A$ as a linear combination of projections into eigenspaces.

The eigenvalues are $\lambda_{1,2}=\pm 2\sqrt{3}$. The eigenvectors are $\begin{pmatrix}-\sqrt{3} \\1\end{pmatrix}$ and $\begin{pmatrix}\frac{1}{\sqrt{3}} \\1\end{pmatrix}$. How can we continue to get the projections?

Answer 1

Hint: Your vectors are orthogonal. Check $P\propto uu^{T}$.