= Ky Fan maximum principle
{c}
For a real <symmetric matrix> $X$ and $1\leq k\leq n$,
$$
\sum_{i=1}^k\lambda_i(X)
=\max_{Q^TQ=I_k}\operatorname{tr}(Q^TXQ).
$$
Indeed $Y=QQ^T$ is a rank-$k$ <orthogonal projection matrix> in the <fantope>, and projecting onto the <eigenvectors> of the largest <eigenvalues> attains the <support function> maximum. Equivalently, the largest <matrix trace> of a compression to a $k$-dimensional <vector subspace> is the <sum of the largest eigenvalues>.
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