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 eX or exp(X), is the n × n matrix given by the power series
e X = ∑ k = 0 ∞ 1 k ! X k {\displaystyle e^{X}=\sum _{k=0}^{\infty }{\frac {1}{k!}}X^{k}}
where X 0 {\displaystyle X^{0}} is defined to be the identity matrix I {\displaystyle I} with the same dimensions as X {\displaystyle X} , and X k = X X k − 1 {\displaystyle X^{k}=XX^{k-1}} for integer k > 0 {\displaystyle k>0} . The series always converges, so the exponential of X is well-defined. Equivalently,
e X = lim k → ∞ ( I + X k ) k {\displaystyle e^{X}=\lim _{k\rightarrow \infty }\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 X t {\displaystyle Y(t)=e^{Xt}} of the (matrix) differential equation
d d t Y ( t ) = X Y ( t ) , Y ( 0 ) = I , {\displaystyle {\frac {d}{dt}}Y(t)=X\,Y(t),\quad 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.
Properties
Elementary properties Let X and Y be n × n complex matrices and let a and b be arbitrary complex numbers. We denote the n × n identity matrix by I and the zero matrix by 0. The matrix exponential satisfies the following properties. We begin with the properties that are immediate consequences of the definition as a power series:
e0 = I exp(XT) = (exp X)T, where XT denotes the transpose of X. exp(X*) = (exp X)*, where X* denotes the conjugate transpose of X. If Y is invertible then eYXY−1 = YeXY−1. If X Y = Y X {\displaystyle XY=YX} then e X e Y = e X + Y {\displaystyle e^{X}e^{Y}=e^{X+Y}} . The proof of this last identity is the same as the standard power-series argument for the corresponding identity for the exponential of real numbers. That is to say, as long as X {\displaystyle X} and Y {\displaystyle Y} commute, it makes no difference to the argument whether X {\displaystyle X} and Y {\displaystyle Y} are numbers or matrices. This identity typically does not hold if X {\displaystyle X} and Y {\displaystyle Y} do not commute (see Golden-Thompson inequality below). Special cases of this identity include:
eaXebX = e(a + b)X eXe−X = I Using the above results, we can easily verify the following claims:
If X is symmetric then eX is also symmetric. If X is skew-symmetric then eX is orthogonal. If X is Hermitian then eX is also Hermitian. If X is skew-Hermitian then eX is unitary. Finally, a Laplace transform of matrix exponentials amounts to the resolvent,
∫ 0 ∞ e − t s e t X d t = ( s I − X ) − 1 {\displaystyle \int _{0}^{\infty }e^{-ts}e^{tX}\,dt=(sI-X)^{-1}}
for all sufficiently large positive values of s.
Linear differential equation systems
One of the reasons for the importance of the matrix exponential is that it can be used to solve systems of linear ordinary differential equations. The solution of
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