Which complex vector multiplied by its conjugate returns the identity matrix

Question

I am trying to find (in case there is any) which complex vector $n$ of 2 dimensions, multiplied by its conjugate transpose, returns a diagonal matrix.

$n = [a, b]^T = [a_1+ja_2, b_1+jb_2]^T$

$nn^\dagger = I$

If I work on the equation, I obtain the following set of 4 equations:

$a_1^2+a_2^2=1$

$b_1^2+b_2^2=1$

$a_1b_1+a_2b_2=0$

$a_1b_2+a_2b_1=0$

But I cannot find the solution to it.

Answer 1

There is no solution. Given a vector $n \in \mathbb{C}^2$, $nn^{\dagger}$ has rank (at most) $1$, whereas the $2 \times 2$ identity matrix has rank $2$. One can produce diagonal matrices this way, but by this consideration, these will always have at most $1$ nonzero entry.