Given:
$$A = \begin{pmatrix} 1 & -i \\ i & 1 \end{pmatrix}$$
Find a unitary matrix $U$ such that $U^{*} A U = D$ where $D$ is diagonal.
Now the eigenvalues of $A$ are $0$ with eigenvector $(i,1)$ and $2$ with eigenvector $(-i,1)$ and they are not orthogonal to one another. So we need to apply gram-schmidt process to orthogonalize this basis.
But once I started, the norm of the vectors is $0$ so I can not normalize them! What do I do in such a case?
Best Answer
The norm of the vectors can't be zero as they both are non-zero vectors. Assuming you're working with the usual euclidean inner (scalar) product, we have:
$$||(i,1)||^2=\langle (i,1)\,,\,(i,1)\rangle=i\cdot\overline i+1\cdot\overline 1=i(-i)+1\cdot 1=2\\||(-i,1)||^2=\langle (-i,1)\,,\,(-i,1)\rangle=(-i)\cdot\overline{(-i)}+1\cdot\overline 1=(-i)(i)+1\cdot 1=2$$
so both vectors' norm is $\;\sqrt2\;$ ...