I understand that the determinant of any unitary matrix is an absolute value of 1.
$|\det(U)|^2 = \overline{\det(U)}\det(U) =\det(U^*)\det(U) = \det(U^* U) = \det(I) = 1$
Is there any unitary matrix that has determinant that is not $\pm 1$ or $\pm i$ ?
For example, is there any unitary matrix such that $\det(U) = \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}} i$…?
I created unitary matrices randomly and computed the determinants but all of them are either $\pm 1$ or $\pm i$
Best Answer
The general expression of a $2\times 2$ unitary matrix is $$ {\displaystyle U={\begin{bmatrix}a&b\\-e^{i\varphi }b^{*}&e^{i\varphi }a^{*}\\\end{bmatrix}},\qquad \left|a\right|^{2}+\left|b\right|^{2}=1,} $$ which depends on 4 real parameters (the phase of $a$, the phase of $b$, the relative magnitude between $a$ and $b$, and the angle $φ$).
The determinant of such a matrix is $$ {\displaystyle \det(U)=e^{i\varphi }.} $$
You could for instance let $\varphi=\pi/4$ to get $\det(U)=\frac{\sqrt 2}{2}+\frac{\sqrt 2}{2}i$.