Computing eigenvalues and eigenvectors I need a little help with this question:
A is this self-adjoint matrix 
$$\pmatrix{ 1 & 0 & i \\
    0 & 2 & 0\\
    −i & 0 & 1}$$
Compute the eigenvalues and a set of orthonormal eigenvectors of A.
I know how to compute the eigenvalues but my eigenvectors seem to be different. They normalise them for some reason
Any help would be appreciated, thanks
 A: To find the eigenvalues you have to solve the equation:
$$\det\pmatrix{ 1-\lambda & 0 & i \\
    0 & 2-\lambda & 0\\
    −i & 0 & 1-\lambda}=0$$
and developing the determinant you find: $\lambda(2-\lambda)(\lambda-2)=0$, so the eigenvalues are: $\lambda_1=0$,$\lambda_2=2$ and $\lambda_2=2$.
The eigenspace of $\lambda_1$ is the subspace of vectors that verify the equation:
$$
\pmatrix{ 1-\lambda_1 & 0 & i \\
    0 & 2-\lambda_1 & 0\\
    −i & 0 & 1-\lambda_1}\pmatrix{x\\y\\z}=\pmatrix{0\\0\\0}
$$
and thay have the form
$$
\pmatrix{-iz\\0\\z}
$$
so you can chose :
$$v_1=
\pmatrix{-i\\0\\1}
$$
In the same way, the eigenspace of $\lambda_2=\lambda_3$ is:
$$
\pmatrix{iz\\y\\z}
$$
and you can chose the two orthogonal vectors:
$$v_2=
\pmatrix{i\\0\\1}
\qquad
v_3=
\pmatrix{0\\1\\0}
$$
Now you can normalize these vectors with: $u_i=\dfrac{v_i}{||v_i||}$ (where the norm is defined as $||v||=\sqrt{\langle v,\bar v\rangle}$ in a complex vector space). So you find:
$$
||v_1||=\sqrt{2}\qquad ||v_2||=\sqrt{2}\qquad ||v_3||=\sqrt{1}
$$
