Nonnegative matrix

Remarks

One writes $A \ge 0$ for a nonnegative matrix and $A > 0$ for a positive matrix. More generally, $A \ge B$ means that $A - B \ge 0$ entry by entry. Example: $\begin{pmatrix} 1 & 2 \\ 2 & 1 \end{pmatrix}$ is positive, hence nonnegative. It is not positive semidefinite, because it has the eigenvalue $-1$. Non-example: $\begin{pmatrix} 1 & -1 \\ 0 & 1 \end{pmatrix}$ is not nonnegative, because its entry in row $1$ and column $2$ is $-1$. Products and sums of nonnegative matrices are nonnegative.