Let:
$$A=\begin{pmatrix}
1&i&i\\
i&1&i\\
i&i&1
\end{pmatrix}$$
I have proved that $A$ is normal. Now I want to find a unitary matrix $P$ such that $P^*AP$ is a diagonal matrix ($P^*$ is the conjugate transpose of $P$).
The eigenvalues of $A$ are: $\{1+2i, 1-i\}$ and the eigenvectors: $\{(1,1,1)^t, (-1,0,1)^t,(-1,1,0)^t\}$
How can I get this matrix $P$?
Best Answer
Choose the eigenvectors corresponding to $1-i$ so they are orthogonal to $(1,1,1)^T$ and each other. For example, replace the third eigenvector by $(1,-2,1)^T$. Then normalize the eigenvectors. This gives: $$P = \pmatrix{ 1 & -1 & 1 \\ 1 & 0 & -2 \\ 1 & 1 & 1 }\pmatrix{\frac{1}{\sqrt{3}} & 0 & 0 \\ 0 & \frac{1}{\sqrt{2}} & 0\\0 & 0 & \frac{1}{\sqrt{6}} }$$ Then $P^* A P$ is diagonal.