Audrina Jackson

2022-07-08

Consider the 4-by-4 matrix $\mathit{M}={\mathit{M}}_{0}+{\mathit{M}}_{1}$, where

${\mathit{M}}_{0}=\alpha \left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& -1& 0\\ 0& 0& 0& -1\end{array}\right)$ and ${\mathit{M}}_{1}=\beta \left(\begin{array}{cccc}0& \gamma & 0& -{\gamma}^{\ast}\\ {\gamma}^{\ast}& 0& -{\gamma}^{\ast}& 0\\ 0& \gamma & 0& -{\gamma}^{\ast}\\ \gamma & 0& -\gamma & 0\end{array}\right)$

where $\alpha $ and $\beta $ are constants and $\gamma ={\gamma}_{x}+i{\gamma}_{y}$ is complex.

Is it possible to unitary transform $\mathit{M}$ into block off-diagonal form ${\mathit{M}}_{B}$?

Namely, I want to find a unitary transform $\mathit{U}$ so that I can write down ${\mathit{M}}_{B}=\mathit{U}\mathit{M}{\mathit{U}}^{\ast}$ (here ${\mathit{U}}^{\ast}$ is the conjugate transpose).

Explicitly, the required block off-diagonal matrix is (in general form)

${\mathit{M}}_{B}=\left(\begin{array}{cc}0& \mathit{Q}\\ {\mathit{Q}}^{\ast}& 0\end{array}\right)$ where $\mathit{Q}=\left(\begin{array}{cc}{Q}_{z}& {Q}_{x}-i{Q}_{y}\\ {Q}_{x}+i{Q}_{y}& -{Q}_{z}\end{array}\right)$

Is there a general recipe to find such a unitary transformation matrix $\mathit{U}$ which leads to the block off-diagonal form, $\mathit{M}\to {\mathit{M}}_{B}$?

${\mathit{M}}_{0}=\alpha \left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& -1& 0\\ 0& 0& 0& -1\end{array}\right)$ and ${\mathit{M}}_{1}=\beta \left(\begin{array}{cccc}0& \gamma & 0& -{\gamma}^{\ast}\\ {\gamma}^{\ast}& 0& -{\gamma}^{\ast}& 0\\ 0& \gamma & 0& -{\gamma}^{\ast}\\ \gamma & 0& -\gamma & 0\end{array}\right)$

where $\alpha $ and $\beta $ are constants and $\gamma ={\gamma}_{x}+i{\gamma}_{y}$ is complex.

Is it possible to unitary transform $\mathit{M}$ into block off-diagonal form ${\mathit{M}}_{B}$?

Namely, I want to find a unitary transform $\mathit{U}$ so that I can write down ${\mathit{M}}_{B}=\mathit{U}\mathit{M}{\mathit{U}}^{\ast}$ (here ${\mathit{U}}^{\ast}$ is the conjugate transpose).

Explicitly, the required block off-diagonal matrix is (in general form)

${\mathit{M}}_{B}=\left(\begin{array}{cc}0& \mathit{Q}\\ {\mathit{Q}}^{\ast}& 0\end{array}\right)$ where $\mathit{Q}=\left(\begin{array}{cc}{Q}_{z}& {Q}_{x}-i{Q}_{y}\\ {Q}_{x}+i{Q}_{y}& -{Q}_{z}\end{array}\right)$

Is there a general recipe to find such a unitary transformation matrix $\mathit{U}$ which leads to the block off-diagonal form, $\mathit{M}\to {\mathit{M}}_{B}$?

torpa6d

Beginner2022-07-09Added 7 answers

I think this $4\times 4$ matrix is the answer.

$\frac{1}{\sqrt{2}}\left(\begin{array}{cccc}1& 0& 1& 0\\ 0& 1& 0& 1\\ -I& 0& I& 0\\ 0& -I& 0& I\end{array}\right)$

$\frac{1}{\sqrt{2}}\left(\begin{array}{cccc}1& 0& 1& 0\\ 0& 1& 0& 1\\ -I& 0& I& 0\\ 0& -I& 0& I\end{array}\right)$

daktielti

Beginner2022-07-10Added 2 answers

What do you mean by "transform"? Are you looking for a similar matrix? If not, what are you trying to do? Are you trying to compute a determinant?

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