Consider a square matrix $A$ with the property $\sum_j A_{ij}=1$ for all $i$. That is, I've relaxed the positive element requirement for a (right) stochastic matrix. What properties do these matrices have?
It appears that provided the eigenvalues $\{\lambda_j\}$ of $A$ all have $|\lambda_j|\leq 1$ a Markov chain consisting of application of $A^k$ to some quasiprobability vector $\vec v$ with $\sum_j v_j=1$ should converge to a stationary distribution.
Do such matrices have a name in the mathematical literature? Are there results on convergence and mixing times of such generalized Markov chains?
You need more than $\ |\lambda_j|\le1\ $ to get convergence of $\ A^k\ $. If $$ A=\pmatrix{1&0&0\\0&-1&2\\ 0&0&1}\ , $$ for instance, the eigenvalues of $\ A\ $ are $\ 1,-1\ $ and $\ 1\ $, but $\ A^k=A\ $ for $\ k\ $ odd and $\ A^k\ $ is the $\ 3\times3\ $ identity matrix for $\ k\ $ even.
Addendum
In general $$ A^k=E\,\pmatrix{\Lambda_1^k&0&\dots&0\\0&\Lambda_2^k&\dots&0\\ \vdots&&\ddots&\vdots\\ 0&0&\dots&\Lambda_r^k}\,E^{-1}\ , $$ where each $\ \Lambda_j\ $ is a Jordan block corresponding to eigenvalue $\ \lambda_j\ $, and $\ E\ $ is a matrix whose columns are generalised eigenvectors of $\ A\ $. The Jordan block $\ \Lambda_j^k\ $ will converge as $\ k\rightarrow\infty\ $ if and only if either:
Therefore, $\ A^k\ $ will converge as $\ k\rightarrow\infty\ $ if and only if every one of its Jordan blocks satisfies one of these two conditions.