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Math Help - problem involving hermitian matrices

  1. #1
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    problem involving hermitian matrices

    Let M_1, M_2, M_3, and M_4 be hermitian matrices that satisfy:

    (M_i)(M_j) + (M_j)(M_i) = 2δ_ij I , for i,j = 1,2,3,4. δ_ij is the kronecker delta symbol and I is the identity matrix.
    a) show that the eigenvalues of M_i are +/- 1.

    It gives a hint to use the eigenbasis of M_i and following the book's advice i find that det(M_i) = +/- 1 which means that the product of its eigenvalues is +/- 1. I also found that Tr((M_i)(M_i)) = n where n is the dimension of the matrix. I also use the fact that if λ is an eigenvalue of A then λ^2 is an eigenvalue of A^2. Since det((M_i)^2) = 1 and Tr((M_i)^2) = n i conclude that the eigenvalues must be +/- 1 because if any of the eigenvalues were fractions then this wouldn't be true.

    b) by considering the relation (M_i)(M_j) = -(M_j)(M_i) when i=/=j show that M_i has no trace.

    this is the part that i am stuck on. i was able to show that Tr((M_i)(M_j)) = 0 but i am stuck after that. the book gave the hint that Tr(ACB) = Tr(CBA) but i haven't been able to figure out how to use that. could someone offer a hint or two? and if its not too much trouble could someone also check my solution to part a) whether my argument was good enough? thanks.
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  2. #2
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    Re: problem involving hermitian matrices

    Solution of a) looks fine. For b), use the fact that (M_iM_j)M_j = (-M_jM_i)M_j and take the trace of both sides.

    (I assume that "M_i has no trace" is just an ungrammatical way of saying "M_i has trace zero". Of course M_i does have a trace, but it is equal to 0.)
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  3. #3
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    Re: problem involving hermitian matrices

    i got it now. thanks for your help!
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