expMij

expMij is a notation used in some mathematical texts to denote a specific entry of the matrix exponential of a matrix M. More precisely, expMij refers to the (i, j) element of the matrix e^M, where e^M is the matrix exponential of M. The matrix exponential is defined by the convergent power series e^M = sum_{k=0}^∞ M^k / k!, and it also arises as the unique solution to the differential equation dX/dt = M X with X(0) = I, evaluated at t = 1.

The full matrix exponential e^M is often computed or approximated, after which the entry expMij is read off as (e^M)_{ij}. When M is diagonalizable, e^M can be computed as V e^D V^{-1} if M = V D V^{-1}; for matrices in Jordan form, e^M is obtained from e^J with Jordan blocks. Practical numerical methods for e^M include scaling and squaring combined with Padé approximants; for large sparse matrices, Krylov subspace methods may be used to approximate specific entries or rows of e^M rather than the full matrix.

Key properties of the matrix exponential underpinning expMij include that e^M is always invertible with inverse e^{-M}, and det(e^M) = e^{tr(M)}. If M is real, e^M is real as well. The (i, j) entry expMij can carry interpretations in various contexts, such as state-transition probabilities in dynamical systems, influence measures in networks, or solutions to linear systems of differential equations.

See also: Matrix exponential, State-transition matrix, Jordan form, Scaling and squaring, Padé approximation.