Spectral Decomposition

[warwick]

[itex]
\begin{bmatrix}
1 & 2 \\
2 & 1 \\
\end{bmatrix}[/itex] = A

(1) Verify that L_A possesses a spectral decomposition.
(2) For each eigenvalue of L_A, explicitly define the orthogonal projection on the corresponding eigenspace.
(3) Verify your results using the spectral theorem.

The eigenvalues of A are 3 and -1. The eigenvectors are (1,1) and (-1,1), respectively. I'm not sure how to proceed.

 

[warwick]

Anyone?