given is
$A = \frac{1}{2} \begin{pmatrix} x & 1 & 1 & 1 \\ y & 1 & -1 & 1 \\ z & 1 & -1 & -1 \\w & 1 & 1 & -1 \end{pmatrix} $
How do I have to chose x,y,z,w in $\mathbb{R}$ so that A is orthogonal and how in $\mathbb{C}$ so that A is unitary matrix?
Is it simply calculating $A^{-1}$ and $A^t$ and so on?
Actually, both rows and columns of a orthogonal matrix form an orthonormal basis. It can clearly be seen that as the magnitude of each element of an orthonormal basis is 1, we have |x|=|y|=|z|=|w|=1. Now you need to determine their signs. To do that, take the dot product of row 1 and row 2. As both are vectors of an orthonormal basis, the dot product is zero. Which means that xy+1=0.=>xy=-1. So x and y are of opposite signs. Similarly find the dot products of rows 2&3 and rows 3&4. You get yz=-1 and zw=-1. So you know that x,z are of same sign and y, w are of same sign. So you have two solutions {x,y,z,w}->{1,-1,1,-1} &{-1,1,-1,1}