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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 $B$ _{so $||B||=\lambda_1(B)$} 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 $B$ real symmetric and $A$ 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).