Finite column generator

Let $S=\{0,1,\dots ,n-1\}$ with $n\ge 1$. Identify its real matrices with $M_n(\mathbb{R})$, the finite matrix space with row and column indices in $S$ from The vector space $M_{m\times n}(F):=F^{m\times n}$ of $m$ by $n$ matrices over a field, with entrywise operations, and use the matrix-product and identity conventions of Rectangular matrix multiplication and the identity matrix $I_n$, including zero-sized shapes. A finite column generator on $S$ is a real matrix $Q=(q_{ij})\in M_n(\mathbb{R})$ satisfying

$q_{ij}\ge 0(i\mathrm{\ne }j),{\sum }_{i<n}{q}_{ij}=0(j<n)\mathrm{.}$

The column index $j$ is the source state and the row index $i$ is the destination state: for $i\mathrm{\ne }j$, $q_{ij}$ is the transition rate from $j$ to $i$. The exit rate from $j$ is

$r_j:=-q_{jj}={\sum }_{i\mathrm{\ne }j}{q}_{ij}\ge 0.$

A column probability vector is $p\in {\mathbb{R}}^n$ with $p_i\ge 0$ for each $i$ and ${\sum }_{i<n}{p}_i=1$. A subprobability vector has the same coordinatewise nonnegativity and has coordinate sum at most one. These conventions are used throughout this page.