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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). The restriction of positive semidefinite $A$ cannot be relaxed.

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 $n\times n$ matrices, is given in The design of suboptimal linear time-varying systems (1968). The inequality still holds for $B$ real symmetric and $A$ positive semidefinite, as proven in Trace bounds on the solution of the algebraic matrix Riccati and Lyapunov equation (1986). The restriction of positive semidefinite $A$ cannot be relaxed, as pointed out in Inequalities for the trace of a matrix product (1994).

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).

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). The restriction of positive semidefinite $A$ cannot be relaxed.