I'm a computer science expert, not a mathematics expert, so, please, be patient for my terminology.
I have a stochastic matrix derived from a Markov Chain. I need to change the probability of one of its rows (with sum of entries always $1$) with a same length vector of probabilities. In particular, I need to proportionally modify the probability of each entry of that line according to the values of the vector. This is an example:
A row of a stochastic matrix is $A=0.2$, $B=0.3$, $C=0.5$ (sum is $1$).
My associated vector is like another stochastic matrix row, so it assigns probability to each term, like $A = 0.2$, $B=0.2$ and $C=0.6$ (sum $1$ again).
How can I proportionally increase or decrease the probability of $A$, $B$ and $C$ in the stochastic matrix according to the vector? The sum of probabilities should always be $1$.
I suppose it is a simpler probability problem than what I presented to you, but I hope that the explanation of the entire scenario helps in understanding the case. Thanks.
The only way to have a sum of elements proportional to the elements of the vector that sum to $1$ is to divide the elements of the vector by their sum.
Example: Vector is $(.3,.4,.5)$ with sum $1.2$ and proportions $3:4:5$ Modified vector is $\left(\frac{.3}{1.2},\frac{.4}{1.2},\frac{.5}{1.2}\right)$ with sum $1$ and proportion $3:4:5.$ The probabilities in the stochastic matrix don't come into it at all.
If this is not what you are looking for, please explain how it violates the conditions you seek.