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A proof of $$\lambda_n(A){\rm tr}\,B\leq {\rm tr}\,AB\leq\lambda_1(A){\rm tr}\,B$$ where $\lambda_n$ is the $n$-th largest eigenvalue of $A$, and $A$, $B$ are positive semidefinite, is given in D.L. Kleinman and M. Athans, "The design of suboptimal linear time-varying systems" (1968). The inequality still holds for $A$ real symmetric and $B$ positive semidefinite, as proven in S.D. Wang, T.S. Kuo, and H.F. Hsu, "Trace bounds on the solution of the algebraic matrix Riccati and Lyapunov equation" (1986).