Consider the 4-by-4 matrix M = M 0 + M 1 , where M 0 = α ( 1 0 0 0 0 1 0 0 0 0 − 1 0 0 0 0 − 1 ) and M 1 = β ( 0 γ 0 − γ ∗ γ ∗ 0 − γ ∗ 0 0 γ 0 − γ ∗ γ 0 − γ 0 ) where α and β are constants and γ = γ x + i γ y is complex.Is it possible to unitary transform M into block off-diagonal form M B ?Namely, I want to find a unitary transform U so that I can write down M B = U M U ∗ (here U ∗ is the conjugate transpose).Explicitly, the required block off-diagonal matrix is (in general form) M B = ( 0 Q Q ∗ 0 ) where Q = ( Q z Q x − i Q y Q x + i Q y − Q z ) Is there a general recipe to find such a unitary transformation matrix U which leads to the block off-diagonal form, M → M B ?

Question

Consider the 4-by-4 matrix   M  =      M    0    +      M    1  , where      M    0    =  α      (                            1                          0                          0                          0                                      0                          1                          0                          0                                      0                          0                          −          1                          0                                      0                          0                          0                          −          1                      )   and       M    1    =  β      (                            0                          γ                          0                          −                      γ                      ∗                                                            γ                      ∗                                    0                          −                      γ                      ∗                                    0                                      0                          γ                          0                          −                      γ                      ∗                                                γ                          0                          −          γ                          0                      )  where   α and   β are constants and   γ  =      γ    x    +  i      γ    y   is complex.Is it possible to unitary transform   M into block off-diagonal form       M    B  ?Namely, I want to find a unitary transform   U so that I can write down       M    B    =  U  M      U      ∗   (here       U      ∗   is the conjugate transpose).Explicitly, the required block off-diagonal matrix is (in general form)      M    B    =      (                            0                          Q                                                  Q                      ∗                                    0                      )   where   Q  =      (                                        Q            z                                                Q            x                    −          i                      Q            y                                                            Q            x                    +          i                      Q            y                                    −                      Q            z                                )  Is there a general recipe to find such a unitary transformation matrix   U which leads to the block off-diagonal form,   M  →      M    B  ?

torpa6d · Accepted Answer

I think this   4  ×  4 matrix is the answer.      1          2            (                            1                          0                          1                          0                                      0                          1                          0                          1                                      −          I                          0                          I                          0                                      0                          −          I                          0                          I                      )

daktielti · Accepted Answer

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

Consider the 4-by-4 matrix <mi mathvariant="bold-italic">M = <mi mathvariant="bold-i

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