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Suppose I have two matrices: $$ A\succeq 0\\ B\succ 0 $$ and I know that $$ \langle v_i,Bv_i\rangle - \lambda_i \geq 0 $$ for every normalized eigenpair $(v_i,\lambda_i)$ of $A$. Is this enough to conclude $$ B-A\succeq 0 $$ i.e., the difference is positive semi-definite? I can't seem to prove this or find a counterexample, but I suspect it is true. Any help would be appreciated.

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This is wrong. Consider a simple example $$ B=\Big(\matrix{1 & 0.9\\ 0.9 & 1 }\Big) $$ $$ A=\Big(\matrix{0.9 & 0\\ 0 & 0.95 }\Big) $$ Here $A$ and $B$ both are strictly positively definite and your condition is satisfied. However $B-A$ is not positively semidefinite.

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