Given A and B positive definite matrices.
Is there an inequality relation between trace(AB) and trace(A+B) ?
2026-03-31 07:57:18.1774943838
relation between trace of product and sum of matrices?
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I don't think there's any interesting linear inequality relating them, just as in the scalar case there's not much relating $xy$ and $x+y$.
There are quadratic inequalities that follow from von Neumann's inequality, such as $$4\mathrm{tr}(AB) < \mathrm{tr}(A+B)^2,$$ since $$4\mathrm{tr}(AB) \leq 4\sum_i \lambda_i \mu_i < 4 \sum_{i,j} \lambda_i\mu_j = \left(\sum_i \lambda_i + \mu_i\right)^2 - \left(\sum_i \lambda_i - \mu_i\right)^2 \leq \mathrm{tr}(A+B)^2$$