I'd like to find the spectral decomposition of $A$:
$A = \begin{pmatrix} 2-i & -1 & 0\\ -1 & 1-i & 1\\ 0 & 1 & 2-i \end{pmatrix}$
i.e. $A=\sum_{i}\lambda_i P_i$ where $P_i$ are the coordinate matrices (in the standard basis) of the corresponding orthogonal transformations in the spectral decomposition of $T_A$ and $\lambda_i$ are the eigenvalues.
I started off by showing that $A$ is normal, piece of cake.
Then found the eigenvalues of $A$, those are: $\lambda_1 = 2-i, \lambda_2 = 3-i, \lambda_3 = -i$. I tried using these known facts from the spectral theorem:
- $A=(2-i)P_1+(3-i)P_2-iP_3$
- $I=P_1+P_2+P_3$
- $\forall i\neq j, P_i P_j=0$
- $P^*_i=P_i$
The only example I have in my book uses these but I couldn't get it to work here. The terms don't cancel out it seems.
What else can I try?