Solution of a) looks fine. For b), use the fact that 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.)
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.
Solution of a) looks fine. For b), use the fact that 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.)