= Hermitian effect variational principle
{c}
{title2=$\max_{0\leq B\leq I,\operatorname{Tr}B=r}\operatorname{Tr}(AB)=\sum_{j<r}\lambda_j$}
For a finite-dimensional <Hermitian operator> $A$ with decreasing <eigenvalues> $\lambda_j$ and integer $0\leq r\leq d$, maximizing $\operatorname{Tr}(AB)$ over <positive contractions> $B$ with <trace> $r$ gives the sum of its leading $r$ <eigenvalues>. In an <orthonormal eigenbasis>, the diagonal entries of $B$ lie in $[0,1]$ and sum to $r$. Moving their weight to the largest <eigenvalues> can only increase the objective. An <orthogonal projection> onto their <eigenvectors> attains the maximum.
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