Is the convex combination of stochastic matrices also stochastic?

Question

Let's say we have two stochastic matrices $S$ and $A$.

How can I show that $M$ is also stochastic when $M$ is given by?:

$M = (1-m)A + mS$

EDIT 1:

$m \in \ \mathbb{R} : 0 < m < 1$

Answer 1

Suppose these matrices are $n\times n$ and let $e\in\mathbb R^n$ be the column vector of $1$'s. Then \begin{align} Me &= ((1-m)A+mS)e\\ &= (1-m)Ae + mSe\\ &= (1-m)e + me\\ &= e, \end{align} so that $M$ is (row)-stochastic.

Answer 2

Well, you have to prove that $M$ satisfies the properties of a stochastic matrix, i.e. you must prove that:

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1.The values of $M$ are positive

This should be fairly simple, since the value of $i$-th row and $j$-th column of $M$ is equal to $$m_{ij}=(1-m)a_{ij}+ms_{ij},$$ where $a_{ij}, s_{ij}$ are the $i$-th row, $j$-th column values of $A$ and $S$, respectively.

The expression for $m_{ij}$ is a sum of positive values (why?), therefore positive.

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2.The rows of $M$ sum to $1$.

This can be done in two ways. One (cleaner) is to show that $Me=e$ when $e$ is the vector composed of only ones. This should be simple since you already know that $Se=e$ and $Ae=e$.

Another way to do it is to write out the sum of the $i$-th row as

$$\sum_{j=1}^n m_{ij}$$

now simplify this as

$$\sum_{j=1}^n m_{ij} = \sum_{j=1}^n ((1-m)a_{ij} + ms_{ij})\\ = (1-m)\sum_{j=1}^n a_{ij} + m\sum_{j=1}^n s_{ij} = (1-m)+m=1$$

Naturally, you need to understand what is happening at each step, but it shouldn't take long.