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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 connection between a matrix Lie algebra and the corresponding Lie group.

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

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

where $X^{0}$ is defined to be the identity matrix $I$ with the same dimensions as $X$ .^[1]

The above series always converges, so the exponential of $X$ is well-defined. If $X$ is a 1×1 matrix the matrix exponential of $X$ is a 1×1 matrix whose single element is the ordinary exponential of the single element of $X$ .

Properties

Elementary properties

Let $X$ and $Y$ be $n \times n$ complex matrices and let $a$ and $b$ be arbitrary complex numbers. We denote the $n \times n$ identity matrix by $I$ and the zero matrix by 0. The matrix exponential satisfies the following properties.^[2]

We begin with the properties that are immediate consequences of the definition as a power series:

$e 0 = I$
$exp(X T) = (exp X) T$ , where $X T$ denotes the transpose of $X$ .
$exp(X *) = (exp X) *$ , where $X *$ denotes the conjugate transpose of $X$ .
If $Y$ is invertible then $e YXY -1 = Ye X Y -1 .$

The next key result is this one:

If $XY=YX$ then $e^{X}e^{Y}=e^{X+Y}$ .

The proof of this 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 $X$ and $Y$ commute, it makes no difference to the argument whether $X$ and $Y$ are numbers or matrices. It is important to note that this identity typically does not hold if $X$ and $Y$ do not commute.

Consequences of the preceding identity are the following:

$e aX e bX = e (a + b) X$
$e X e - X = I$

Using the above results, we can easily verify the following claims. If $X$ is symmetric then $e X$ is also symmetric, and if $X$ is skew-symmetric then $e X$ is orthogonal. If $X$ is Hermitian then $e X$ is also Hermitian, and if $X$ is skew-Hermitian then $e X$ is unitary.

Finally, we have the following:

A Laplace transform of matrix exponentials amounts to the resolvent,

\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

{\frac {d}{dt}}y(t)=Ay(t),\quad y(0)=y_{0},

where $A$ is a constant matrix, is given by

y(t)=e^{At}y_{0}.\,

The matrix exponential can also be used to solve the inhomogeneous equation

{\frac {d}{dt}}y(t)=Ay(t)+z(t),\quad y(0)=y_{0}.

See the section on applications below for examples.

There is no closed-form solution for differential equations of the form

{\frac {d}{dt}}y(t)=A(t)\,y(t),\quad y(0)=y_{0},

where $A$ is not constant, but the Magnus series gives the solution as an infinite sum.

The determinant of the matrix exponential

By Jacobi's formula, for any complex square matrix the following trace identity holds:^[3]

$\det(e^{A})=e^{\operatorname {tr} (A)}~.$

An explicit basis in the fundamental, 3, representation can be constructed in analogy to the Pauli matrix algebra of the spin operators. It consists of the Gell-Mann matrices,

{\begin{array}{ccc}\lambda _{1}={\begin{pmatrix}0&1&0\\1&0&0\\0&0&0\end{pmatrix}}&\lambda _{2}={\begin{pmatrix}0&-i&0\\i&0&0\\0&0&0\end{pmatrix}}&\lambda _{3}={\begin{pmatrix}1&0&0\\0&-1&0\\0&0&0\end{pmatrix}}\\\\\lambda _{4}={\begin{pmatrix}0&0&1\\0&0&0\\1&0&0\end{pmatrix}}&\lambda _{5}={\begin{pmatrix}0&0&-i\\0&0&0\\i&0&0\end{pmatrix}}\\\\\lambda _{6}={\begin{pmatrix}0&0&0\\0&0&1\\0&1&0\end{pmatrix}}&\lambda _{7}={\begin{pmatrix}0&0&0\\0&0&-i\\0&i&0\end{pmatrix}}&\lambda _{8}={\frac {1}{\sqrt {3}}}{\begin{pmatrix}1&0&0\\0&1&0\\0&0&-2\end{pmatrix}}.\end{array}}

These are the generators of the $SU(3)$ group in the triplet representation, and they are normalized as

\operatorname {tr} (\lambda _{j}\lambda _{k})=2\delta _{jk}.

The Lie algebra structure constants of the group are given by the commutators of $\lambda _{k}$

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle [\lambda_j,\lambda_k]=2if_{jkl}\lambda_l ~,}

where $f_{jkl}$ are the structure constants completely antisymmetric and are analogous to the Levi-Civita symbol $\epsilon _{jkl}$ of $SU(2)$ .

In general, they vanish, unless they contain an odd number of indices from the set {2,5,7}, corresponding to the antisymmetric $λ$ s. Note $f_{ljk}={\frac {-i}{2}}\mathrm {tr} ([\lambda _{l},\lambda _{j}]\lambda _{k})$ .