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Matrix Exponential ○◂｜Definition｜1st｜20251122201720-00-⌔

Matrix exponential - Wikipedia

Matrix exponential

In mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function. It is used to solve systems of linear differential equations. In the theory of Lie groups, the matrix exponential gives the exponential map between a matrix Lie algebra and the corresponding Lie group.

Let X be an n × n real or complex matrix. The exponential of X, denoted by e${}^X$ or exp(X), is the n × n matrix given by the power series

$e^X=\sum _{k=0}^{\infty }\frac{1}{k!}{X}^k$

where $X^0$ is defined to be the identity matrix $I$ with the same dimensions as $X$, and ⁠ $X^k=X{X}^{k-1}$ ⁠ for integer ⁠ $k>0$ ⁠.¹ The series always converges, so the exponential of X is well-defined.

Equivalently,

$e^X=\underset{k\to \infty }{\lim }{\left(I+\frac{X}{k}\right)}^k$

for integer-valued k, where I is the n × n identity matrix.

Equivalently, the matrix exponential is provided by the solution $Y(t)=e^{Xt}$ of the (matrix) differential equation

$\frac{d}{dt}Y(t)=XY(t),Y(0)=I,$

evaluated at t = 1.

When X is an n × n diagonal matrix then exp(X) will be an n × n diagonal matrix with each diagonal element equal to the ordinary exponential applied to the corresponding diagonal element of X.

Printed 2026-06-28.

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Footnotes

1. Hall 2015 Equation 2.1 ↩

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