Can the trace of a positive matrix increase under a projection?

Consider the following orthogonal basis of $\mathbb S_2$ with respect to the Frobenius inner product: $$ A=\pmatrix{4&0\\ 0&2},\ B=\pmatrix{1&0\\ 0&-2},\ C=\pmatrix{0&1\\ 1&0}. $$ Let $P$ be the orthogonal projection onto $\mathcal V=\operatorname{span}(A)$ and let $X=A+B=\operatorname{diag}(3,0)$. Then $$ \operatorname{tr}(P(X)) =\operatorname{tr}(P(A+B)) =\operatorname{tr}(A) =6>3 =\operatorname{tr}(X). $$

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Linear Algebra

Matrices

Matrix Analysis

Symmetric Matrices

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