TITLE: Trace of non-commutable matrices
QUESTION [11 upvotes]: Let $M_1$ and $M_2$ be two symmetric $d\times d$ matrices. What is the relationship between
$tr(M_1M_2M_1M_2)$ and $tr(M_1^2 M_2^2 )$?
P.S. I tried a few examples and found
$$
tr(M_1M_2M_1M_2) \le tr(M_1^2 M_2^2 )
$$
seems always true. Is there a theorem?

REPLY [10 votes]: Not to take anything away from Suvrit's answer, but this is actually much simpler. First, we can assume $M_1$ is diagonal. Call it $diag(x_1, \dotsc, x_i).$
Then the difference between the LHS and the RHS is 
$$\sum_{i> j} a_{ij}^2 (x_i - x_j)^2,$$
where the $a_{ij}$ are the entries of $M_2.$