When is this matrix unitary

Question

If we have the matrix $$U=\begin{bmatrix} a & b & \frac{1}{\sqrt{2}} \\ c & 0 & 0 \\ d & e &\frac{-1}{\sqrt{2}} \end{bmatrix}$$ what are the conditions on $a,b,c,d,e$ such that the matrix $U$ is unitary. Now I know that unitary means $U^*U=I$, and you could find what $U^*$ is and match it, but I have no clue what to do when I am matching. Also is there an easier way to find the conditions?

Answer 1

Compute the determinant by expansion by minors along the second row. You get that you need

$${1\over\sqrt{2}}|c(b+e)|=1$$

Now add orthonormality of the columns and of the dual matrix, and you get $|c|=1$ and $a=d=0$ and $|e|^2=|b|^2={1\over 2}$. But then we have

$|e+b|\le |e|+|b|=\sqrt 2$

And since equality holds, it must be that $e=b$ since they have the same absolute value.

So the matrix must be of the form

$$\begin{pmatrix} 0 & {1\over\sqrt 2}e^{i\theta} & {1\over\sqrt 2} \\ e^{i\phi} & 0 & 0 \\ 0 & {1\over\sqrt 2}e^{i\theta} & -{1\over\sqrt 2}\end{pmatrix}$$