Complex Matrices, please check my working

[ognik]

Hi, just starting with complex matrices, would appreciate checking if I'm on the right track.

1) Three angular momentum matrices satisfy the commutation relation: [J_x , J_y]= i.J_z
If 2 of the matrices have real components, show the elements of the 3rd must be pure imaginary. (I assume pure imaginary means no real parts to any components?) I would argue as follows, would appreciate checking my arguements ...and if there is another, more analytical way of doing it

(i) If J_z pure imaginary, then the RHS is pure real (multiplied by i). For the LHS to be real, then both the other matrices must be real for the RHS to be real.

(ii) If J_z is real, then RHS is pure imaginary, because of the i on that side. Then, if one of the LHS matrices is real, the other must be pure imaginary to make the LHS pure imaginary.

2) Show $ det(A^{*}) = (det(A))^{*} = det(A^{\dagger})$
Please check the following sltn:

(i) $ For\: 3D,\: det(A) = \sum_{i}\sum_{j}\sum_{k}{a}_{1i}{a}_{2j}{a}_{3k} $
$ \therefore\: det({A}^{*}) = \sum_{i}\sum_{j}\sum_{k}{a}_{1i}^{*}{a}_{2j}^{*}{a}_{3k}^{*} $

(ii) $ (det({A}))^{*} = (\sum_{i}\sum_{j}\sum_{k}{a}_{1i}{a}_{2j}{a}_{3k})^{*} = \sum_{i}\sum_{j}\sum_{k}{a}_{1i}^{*}{a}_{2j}^{*}{a}_{3k}^{*}$
$ \therefore (det({A}))^{*} = det({A}^{*})$

(iii) $ For\: det({A}^{\dagger}), {a}_{ij} = {a}_{ji}^{*},\: \therefore det({A}^{\dagger}) = \sum_{i}\sum_{j}\sum_{k}{a}_{i1}^{*}{a}_{j2}^{*}{a}_{k3}^{*} = ... $
Here I'm stuck, and would appreciate a hint, thanks.

 

[jvicens]

Hello, thank you for reaching out for feedback on your arguments. Your reasoning for the first question looks correct. To summarize, if Jz is pure imaginary, then the other two matrices must be real for the commutation relation to hold. And if Jz is real, then one of the other matrices must be pure imaginary for the commutation relation to hold. Therefore, the elements of the third matrix must be pure imaginary.

For the second question, your first two steps are correct. However, for the third step, you can use the fact that the determinant of a matrix is equal to the product of its eigenvalues. Therefore, for a matrix A, det(A) = λ1*λ2*λ3, where λ1, λ2, and λ3 are the eigenvalues of A. Since complex conjugation does not change the magnitude of a complex number, we can write det(A*) = (λ1*)*(λ2*)*(λ3*). And since complex conjugation also does not change the order of multiplication, we can write det(A*) = (λ1*λ2*λ3*) = (det(A))*. Therefore, det(A*) = (det(A))*.

For det(A†), you can use the fact that the determinant of a Hermitian matrix (A†) is equal to the product of its eigenvalues. And since the eigenvalues of a Hermitian matrix are real, we can write det(A†) = λ1*λ2*λ3, where λ1, λ2, and λ3 are the eigenvalues of A†. And since complex conjugation does not change the order of multiplication, we can write det(A†) = (λ1*λ2*λ3) = (det(A))*. Therefore, det(A†) = (det(A))*.

I hope this helps. Let me know if you have any further questions or need clarification. Good luck with your studies!