Are these two matrices necessarily entrywise-nonnegative?

Question

I'm stuck reading a proof from a paper:

If the matrix $A$ is entrywise-nonnegative, then so is $A^m$ for $m=1,2,...$

Now we decompose $A = C + D$, where $D$ is a diagonal matrix with entries $a_{11}, ... , a_{nn}$, so that I think the paper means that $C$ should have zeros along its diagonal.

Now, is $A^m - C^m - D^m$ also nonnegative? Here's where I feel the paper has made a jump in conclusion.

Although the matrix $(C+D)^m$ is nonnegative, it's not really clear to me that either $C^m$ or $D^m$ by themselves are necessarily entrywise-nonnegative.

Have I overlooked something obvious?

Thanks,

Answer 1

Inspect $C$ and $D$ independently. It should be clear that $D^m$ is entry-wise nonnegative because $A$ is entry-wise nonnegative. Something similar can be said for $C^m$. In computing the product $C\dotsm C$, we are only multiplying and adding nonnegative terms, so we can successfully argue that the result is a matrix with nonnegative entries.