Trace Inequalities for Positive Semidefinite Matrices

In the setting of Symmetric and Positive Semidefinite Matrices, the Operator Norm and Spectral Notation, let $n\in\mathbb{N}$, let $P\in\mathcal{S}(n)$ be positive semidefinite, and let $X\in\mathcal{S}(n)$. Write $\operatorname{tr}$ for the trace and $\lambda_{1}(X)\ge\dots\ge\lambda_{n}(X)$ for the ordered eigenvalues of $X$.

(i) Nonnegativity. If $X$ is positive semidefinite, then $\operatorname{tr}(PX)\ge0$.

(ii) Monotonicity. If $Y\in\mathcal{S}(n)$ and $X\preceq Y$, then $\operatorname{tr}(PX)\le\operatorname{tr}(PY)$.

(iii) Eigenvalue bounds.

$$\lambda_{n}(X)\operatorname{tr}(P)\le\operatorname{tr}(PX)\le\lambda_{1}(X)\operatorname{tr}(P).$$