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

Audrina Jackson 2022-07-08 Answered
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 ?
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Answers (2)

torpa6d
Answered 2022-07-09 Author has 7 answers
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 )
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daktielti
Answered 2022-07-10 Author has 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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