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Let's consider a symmetric matrix A.

If for each row, the diagonal entry is equal or larger than the magnitude of any other element, that is

$$a_{ii} \geq |a_{ij}| \quad\text{for all rows } i \text{ and entries } j , \,$$

then the matrix is positive semi-definite

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It's false, I've just this counterexample.

$$\begin{pmatrix} 1 & 0.9 & 0.9 \\ 0.9 & 1 & 0.1 \\ 0.9 & 0.1 & 1 \end{pmatrix}$$ is indefinite, since the eigenvalues are $0.9$ and $(21 \pm \sqrt{649})/20$.

From this answer: Is this a positive semi- definite matrix

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  • $\begingroup$ Rounding off to integers looks more beautiful (with only zeros and ones). $\endgroup$ – A.Γ. Jan 14 at 10:19

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