expm

expM refers to the matrix exponential, the exponential of a square matrix M. It generalizes the scalar exponential and is defined by the convergent power series exp(M) = sum_{k=0}^∞ M^k / k!. It can also be characterized as the unique matrix that satisfies the differential equation d/dt X(t) = M X(t) with X(0) = I, in which case X(t) = exp(tM).

When M is diagonalizable as M = P D P^{-1}, the matrix exponential is exp(M) = P exp(D) P^{-1}, where exp(D) is the diagonal matrix with entries exp(d_i). In the general case, exp(M) is computed via Jordan form or other stable algorithms; if J is the Jordan form of M, then exp(M) = P exp(J) P^{-1}, and exp(J) is blockwise computed from Jordan blocks.

Key properties include exp(0) = I and the eigenvalues of exp(M) being the exponentials of the eigenvalues of M. In general, exp(M1 + M2) ≠ exp(M1) exp(M2) unless M1 and M2 commute. The matrix exponential is widely used to solve linear systems of ordinary differential equations, assess stability, and describe continuous-time Markov processes among other applications.

Numerical computation often employs Padé approximants with scaling and squaring, or Krylov subspace methods for large sparse matrices. In software, common implementations include MATLAB's expm and SciPy's scipy.linalg.expm, among others. The matrix exponential is closely related to the matrix logarithm and to concepts such as the Jordan form and spectral decomposition.