Is this a positive semi- definite matrix I have a matrix $A$, which satisfies : * $A$ is symmetric; * all the diagonal entries of $A$ are equal to $1$; * other entries of $A$ is between $0$ and $1$. My question is, whether $A$ is a positive semi-definite matrix?

Here's a counterexample for size $3 \times 3$ (there are probably simpler examples, but well...): $$\begin{pmatrix} 1 & 0.9 & 0.9 \\\ 0.9 & 1 & 0.1 \\\ 0.9 & 0.1 & 1 \end{pmatrix}$$ is indefinite, since the eigenvalues are $0.9$ and $(21 \pm \sqrt{649})/20$.