Let $K\_1$ and $K\_2$ be two $N \times N$ stochastic matrices (hence non-negative and rows adding up to one) with zeros on the diagonal. If $\alpha \in (0,1)$, is it possible to have 


$$K\_1 K\_2 = \alpha K\_1 + (1-\alpha)K\_2?$$



Product and convex combination of two stochastic matrices
Let $$A=\pmatrix{0&.5&0&0&.5\cr.5&0&.5&0&0\cr0&.5&0&.5&0\cr0&0&.5&0&.5\cr.5&0&0&.5&0\cr}{\rm\ and\ }B=\pmatrix{0&0&.5&.5&0\cr0&0&0&.5&.5\cr.5&0&0&0&.5\cr.5&.5&0&0&0\cr0&.5&.5&0&0\cr}$$ Then $AB=.5A+.5B$. 


