TITLE: Ensuring that a symmetric matrix with nonnegative elements is positive semidefinite
QUESTION [6 upvotes]: I have the following matrix $A$:

symmetric
all positive and/or zero values
the main diagonal is all the same value, $x$.

To ensure that the matrix $A$, is positive semidefinite, must I only ensure that $x \geq 0$?  It seems correct from my thinking, but wanted to make sure. Thanks.

REPLY [7 votes]: Note however that a diagonally dominant symmetric matrix is positive semi definite.
In your case it is sufficient to add the condition $x \ge \sum_{j\neq i} A_{ij}, i\neq j$ to ensure $A$ to be positive semi definite.