expSigmaij

expSigmaij is a notation encountered in mathematics and related fields. It is not a single universal object but a context-dependent shorthand that can denote either the (i,j) entry of the matrix exponential of a square matrix Sigma with entries sigma_{kl}, or, in contexts where Sigma_{ij} is a scalar, the scalar exponential e^{Sigma_{ij}}.

In general, the matrix exponential exp(Sigma) is defined by the convergent series exp(Sigma) = I + Sigma + (1/2) Sigma^2 + (1/6) Sigma^3 + ... . The (i,j) entry of exp(Sigma) is denoted [exp(Sigma)]_{ij}, and some authors write expSigmaij to refer specifically to this entry.

When Sigma is diagonal, exp(Sigma) is diagonal with entries e^{sigma_{ii}}; off-diagonal entries vanish. For non-diagonal Sigma, exp(Sigma) is typically dense, reflecting interactions among components.

Common uses include solving linear dynamical systems and studying state propagation. For dx/dt = Sigma x, the solution is x(t) = exp(t Sigma) x(0); the matrix exp(t Sigma) describes how initial components influence future states, with [exp(t Sigma)]_{ij} indicating the influence from component j to i over time t. In probability, exp(t Q) with Q a generator matrix yields transition probabilities in continuous-time Markov chains, where [exp(t Q)]_{ij} gives the probability of moving from j to i in time t.

Caution: some writers use expSigmaij to mean e^{Sigma_{ij}} when Sigma_{ij} is scalar, or to denote the (i,j) entry of exp(Sigma); context should determine the intended meaning.

See also: matrix exponential; generator matrix; Markov chain; covariance matrix.