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The Wikipedia page for unitary matrices gives a general expression for the $2 \times 2$ unitary matrix:

\begin{pmatrix} a & b \\ -e^{i\phi}b^* & e^{i\phi}a^* \end{pmatrix}

with $|a^2| + |b^2| = 1$. Is there a similar general construction for the $3 \times 3$ unitary matrix?

Rodrigo de Azevedo
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Thomas Markov
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  • It seems unlikely; 2x2 orthogonal (real) matrices have a similarly nice form, but there's not really any such for the 3x3 because the shape is just substantially more complicated than $\mathcal{S}^1$, and I can't imagine that 'complexifying' them changes that. – Steven Stadnicki Jun 19 '17 at 16:22
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    One can simplify this question a bit by focusing only on special unitary matrices where $\det{U}=1$. In that case, the construction for $SU(3)$ is relevant in physics and is discussed in this question on Physics.SE: https://physics.stackexchange.com/q/237988/55641. – Semiclassical Jun 19 '17 at 17:04
  • (One thing to note: the top answer on the physics.SE question gives what amounts to an 'Euler angles' parametrization, and that parametrization has problems at its boundaries - in particular, it's not one-to-one (or even consistently n-to-1) without some rather unnatural restrictions on its parameters.) – Steven Stadnicki Jun 19 '17 at 18:29

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I have not checked this paper myself, but an answer is probably contained in the rather recent paper:

Gil, J. J. (2018). Parametrization of 3× 3 unitary matrices based on polarization algebra. The European Physical Journal Plus, 133(5), 206.

Parametrization of unitary 3x3 matrix

Matteo
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