the Q-matrix (rate matrix)
The Q-matrix, also called the generator or rate matrix, is the single object that encodes everything about a continuous-time Markov chain: it lists, for every ordered pair of distinct states (i, j), the rate q_ij at which the chain attempts to jump from i to j. It plays the role that the transition matrix plays for a discrete chain, but in continuous time the natural local description is rates, not one-step probabilities. The generator is the infinitesimal version of the dynamics — the derivative of the transition semigroup at time zero.
A matrix Q = (q_ij) on a countable state space is a Q-matrix (a conservative generator) when its off-diagonal entries are nonnegative, q_ij >= 0 for i != j, and each row sums to zero, q_ii = -sum_{j != i} q_ij <= 0. The off-diagonal q_ij is the rate of the i->j transition; q_i = -q_ii is the total rate of leaving i, i.e. the holding rate. The connection to probabilities is infinitesimal: P_h(i,j) = q_ij h + o(h) for i != j, and P_h(i,i) = 1 + q_ii h + o(h) as h -> 0, so Q = lim_{h->0} (P_h - I)/h = P'_0. This is exactly the statement that the generator is the time-derivative of P_t at 0. From Q one recovers everything: the holding rates q_i = -q_ii, the jump-chain kernel K(i,j) = q_ij/q_i, the forward and backward Kolmogorov equations P'_t = P_t Q = Q P_t, and (when the chain is non-explosive) the transition function P_t = e^(tQ).
The generator is the right level of description because rates compose by addition while probabilities compose by multiplication: superposing independent transition mechanisms just adds their Q-matrices. This is why reaction networks, queueing networks and population models are specified by listing rates. A subtlety on infinite state spaces: row sums equal to zero (conservativeness) does NOT guarantee the chain is well-behaved. If the holding rates q_i grow too fast the minimal process built from Q can explode, and then P_t = e^(tQ) must be interpreted carefully — the naive matrix exponential of an unbounded operator need not be a stochastic (rows-sum-to-one) matrix. For unbounded generators e^(tQ) is a genuine operator-theory statement, not a convergent power series term by term.
A Poisson process of rate lambda, viewed as a CTMC on {0,1,2,...} that only goes up, has Q with q_{n,n+1} = lambda, q_{n,n} = -lambda, and all other entries 0 — an upper-bidiagonal generator. Its forward equation P'_t = P_t Q reproduces exactly the Poisson distribution P_t(0,n) = e^(-lambda t) (lambda t)^n / n!.
Reading a CTMC's dynamics straight off its generator: off-diagonal entries are jump rates, the diagonal is minus the total exit rate.
Conservativeness (rows summing to zero) is necessary but not sufficient for good behavior on an infinite state space: the chain may still explode if the exit rates q_i grow too fast, in which case e^(tQ) need not be a stochastic matrix and the generator must be treated as an unbounded operator.