# Product of two matrices of convex combinations [closed]

How to show that product of two matrices $\mathbf{A}$ and $\mathbf{B}$ of convex combinations as $\mathbf{C=A*B}$ is also a matrix of convex combinations.

Convex combinations: entries of each column of matrix are non-negative and they sum to 1.

• "Convex combination" is what's usually called "stochastic", right? Commented Jan 8, 2018 at 16:37
• @GerryMyerson I guess the word "stochastic" is used in reference to a square probability/transition matrix. In my example none of the matrices are square. I would probably use the term "convex combination" or a combination that is "conic & affine". May be I am not missing anything? Commented Jan 9, 2018 at 5:11
• OK, I guess I assumed the matrices were square. Anyway, Sebastian seems to have settled things. Commented Jan 11, 2018 at 9:33

Just turn being a convex combination into vector arithmetic: $(1, \ldots, 1) \cdot C = (1, \ldots, 1) \cdot A \cdot B = (1, \ldots, 1) \cdot B = (1, \ldots, 1)$. The non-negativity is clear I suppose.