Matrix exponential

The exponential of a square matrix, defined by the convergent power series exp(A) = I + A + A²/2! + ···. It generalizes scalar exponentiation and describes solutions to linear systems with constant coefficients.

Matrix power series: A series whose terms are matrices, with convergence defined through matrix norms or equivalent finite-dimensional criteria. The defining series for the matrix exponential is a matrix power series.
Linear differential equation: A differential equation in which the unknown function and its derivatives occur linearly. Constant-coefficient systems are solved directly using the matrix exponential.
Square matrix: A matrix with the same number of rows and columns, allowing powers and determinant operations. The matrix exponential is defined for square matrices because its powers must be composable.
Elementwise exponential: The operation that applies the scalar exponential separately to each matrix entry. It is not the power-series matrix exponential, which depends on matrix multiplication.
Jordan normal form: A canonical form that represents a complex square matrix using Jordan blocks. Jordan blocks make the exponential computable even when a matrix is not diagonalizable.
State-transition matrix: A matrix that maps a system's state at one time to its state at another. For a time-invariant linear system, this matrix is exp(A times elapsed time).
Identity matrix: A square matrix that leaves every compatible vector unchanged under multiplication. It is the constant term in the power series and the exponential's value at zero.
Hadamard product: The entrywise product of two matrices of the same dimensions. Entrywise multiplication defines a different algebra from the one used in the matrix exponential.
Diagonalization: The representation of a square matrix as a diagonal matrix in a suitable basis, when such a basis exists. For diagonalizable matrices, exponentiation reduces to exponentiating eigenvalues.
Continuous-time Markov chain: A stochastic process on discrete states whose transitions occur continuously in time. Its transition probabilities are obtained by exponentiating the rate matrix.

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