A proof of
$$\lambda_n(B)\,{\rm tr}\,A\leq {\rm tr}\,(AB)\leq\lambda_1(B)\,{\rm tr}\,A$$
where $\lambda_n$ is the $n$-th largest eigenvalue of $B$ <sub>so $||B||=\lambda_1(B)$</sub> and $A$, $B$ are positive semidefinite $n\times n$ matrices, is given in <A HREF="http://ieeexplore.ieee.org/document/1098852/">The design of suboptimal linear time-varying systems</A> (1968). The inequality still holds for $B$ real symmetric and $A$ positive semidefinite, as proven in <A HREF="http://ieeexplore.ieee.org/document/1104370/">Trace bounds on the solution of the algebraic matrix Riccati and Lyapunov equation</A> (1986). The restriction of positive semidefinite $A$ cannot be relaxed, as pointed out in <A HREF="http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.636.8021&rep=rep1&type=pdf">Inequalities for the trace of a matrix product</A> (1994).