Given that $A$ is an $n\times n$, symmetric and positive semidefinite, how would you prove that the sum of the entries of $A$ is positive?
2026-03-27 06:12:34.1774591954
Is the sum of the entries of a symmetric positive semidefinite matrix positive?
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Let $X=(1,\ldots,1)\in\mathbb{R}^n$, then since $A$ is symmetric and positive semidefinite, one has : $$XA{}^tX\geqslant 0.$$ On the other hand, one has: $$XA^tX=\sum_{i=1}^n\sum_{j=1}^na_{i,j}.$$ Whence the result.