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I was trying to find the most general way of writing a $4\times 4$ unitary matrix but I got stuck when I obtained a set of equations that did not look very pretty after starting from something like: $$U= \begin{pmatrix} a & b & c & d\\ e & f & g & h\\ i & j & k & l\\ m & n & o & p\\ \end{pmatrix}.$$

I was wondering if there was any clever way to write down such a matrix using a minimal amount of variables, probably under the form of exponentials and cosines/sines. Example of what I am looking for:

$$U=\frac{1}{\sqrt{2}} \begin{pmatrix} {e}^{+i\varphi} & {e}^{+i\varphi} & 0 & 0\\ {e}^{-i\varphi} & -{e}^{-i\varphi} & 0 & 0\\ 0 & 0 & {e}^{+i\psi} & {e}^{+i\psi}\\ 0 & 0 & {e}^{-i\psi} & -{e}^{-i\psi}\\ \end{pmatrix},$$

but without any $0$ unless it is required due to the unitarity of the matrix. I have not found anything on the internet related to this topic, except these threads:

I was hoping for either a systematic method to deal with $n\times n$ unitary matrices or a specific method to the $4 \times 4$ case.

Rodrigo de Azevedo
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Amentia
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2 Answers2

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The columns of your matrix form an orthonormal basis of $\mathbb C^4$. You can use the Gram-Schmidt procedure to start with four linearly independent column vectors $x_1, x_2, x_3, x_4$ and produce an orthonormal basis:

$$ \eqalign{u_1 &= x_1/\|x_1\|\cr u_2 &= (x_2 - u_1 u_1^* x_2)/\|(x_2 - u_1 u_1^* x_2)\|\cr u_3 &= (x_3 - u_1 u_1^* x_3 - u_2 u_2^* x_3)/\| (x_3 - u_1 u_1^* x_3 - u_2 u_2^* x_3)\|\cr u_4 &= (x_4 - u_1 u_1^* x_4 - u_2 u_2^* x_4 - u_3 u_3^* x_4)/\|x_4 - u_1 u_1^* x_4 - u_2 u_2^* x_4 - u_3 u_3^* x_4\|\cr}$$

Robert Israel
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  • Thank you for your answer. I think I understand the idea but I do not see how it solves my problem because I did not get how we choose the vectors $x_{1},x_{2},x_{3},x_{4}$. Even for the case $n=2$, I could start with $x_{1}=(1,1)$ and $x_{2}=(-1,1)$. I think that would work but it would not be the most general form which is proposed as $x_{1}=(a,b)$ and $x_{2}=(-e^{i\phi}b^{},e^{i\phi}a^{})$ on the wikipedia page: https://en.wikipedia.org/wiki/Unitary_matrix – Amentia Dec 29 '23 at 16:14
  • The $x_1$ are arbitrary, as long as they are linearly independent. Of course this isn't a parametrization, because the map from $(x_1, ]ldots, x_4)$ to $(u_1, \ldots, u_4)$ is not one-to-one. You can (almost) make a parametrization by taking $x_1$, the first $3$ elements of $x_2$, and the first $2$ elements of $x_3$, all with norm $1$. The other elements are then chosen (if possible) to make the vectors orthogonal, then the vectors are normalized. – Robert Israel Dec 31 '23 at 22:00
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Hint You could perhaps attempt to write $U$ as a product of matrices having the form $M_k = (I - 2 c_k c_k^*)$ where $c_k$ is a unitary column vector which $k$ first coordinates may be non-zero.

More specifically, you could choose $c_d$ by requiring that $M_d U e_d = e_d$ in dimension $d$, etc, where $e_d$ is the $d$-th basis vector.

Gribouillis
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  • Thank you for your answer. Can you elaborate on why you choose to use this method? The form of your matrix $M_{k}$ looks similar to the Gram-Schmidt procedure proposed in Robert Israel's answer but I do not know if they are actually related or totally different methods. I could already find linearly independent vectors by looking at the wikipedia page on quaternions and 4D rotations but it seems to work only for real vectors... – Amentia Dec 31 '23 at 12:21
  • Is it related to this: https://en.wikipedia.org/wiki/Householder_transformation ? – Amentia Dec 31 '23 at 12:35