Editing Matrix exponential (section)

=== Elementary properties ===
Let {{math|''X''}} and {{math|''Y''}}  be {{math|''n'' × ''n''}} complex matrices and let {{math|''a''}} and {{math|''b''}} be arbitrary complex numbers. We denote the {{math|''n'' × ''n''}} [[identity matrix]] by {{math|''I''}} and the [[zero matrix]] by 0. The matrix exponential satisfies the following properties.<ref>{{harvnb|Hall|2015}} Proposition 2.3</ref>

We begin with the properties that are immediate consequences of the definition as a power series:
* {{math|1=''e''<sup>0</sup> = ''I''}}
* {{math|1=exp(''X''<sup>T</sup>) = (exp ''X'')<sup>T</sup>}}, where {{math|''X''<sup>T</sup>}} denotes the [[transpose]] of {{math|''X''}}.
* {{math|1=exp(''X''<sup>∗</sup>) = (exp ''X'')<sup>∗</sup>}}, where {{math|''X''<sup>∗</sup>}} denotes the [[conjugate transpose]] of {{math|''X''}}.
* If {{math|''Y''}} is [[invertible matrix|invertible]] then {{math|1=''e''<sup>''YXY''<sup>−1</sup></sup> = ''Ye''<sup>''X''</sup>''Y''<sup>−1</sup>.}}

The next key result is this one:
* If <math>XY=YX</math> then <math>e^Xe^Y=e^{X+Y}</math>.

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 <math>X</math> and <math>Y</math> commute'', it makes no difference to the argument whether <math>X</math> and <math>Y</math> are numbers or matrices. It is important to note that this identity typically does not hold if <math>X</math> and <math>Y</math> do not commute (see [[#Inequalities for exponentials of Hermitian matrices|Golden-Thompson inequality]] below).

Consequences of the preceding identity are the following:
* {{math|1=''e''<sup>''aX''</sup>''e''<sup>''bX''</sup> = ''e''<sup>(''a'' + ''b'')''X''</sup>}}
* {{math|1=''e''<sup>''X''</sup>''e''<sup>−''X''</sup> = ''I''}}

Using the above results, we can easily verify the following claims. If {{math|''X''}} is [[symmetric matrix|symmetric]] then {{math|''e''<sup>''X''</sup>}} is also symmetric, and if {{math|''X''}} is [[skew-symmetric matrix|skew-symmetric]] then {{math|''e''<sup>''X''</sup>}} is [[orthogonal matrix|orthogonal]]. If {{math|''X''}} is [[Hermitian matrix|Hermitian]] then {{math|''e''<sup>''X''</sup>}} is also Hermitian, and if {{math|''X''}} is [[skew-Hermitian matrix|skew-Hermitian]] then {{math|''e''<sup>''X''</sup>}} is [[unitary matrix|unitary]].

Finally, a [[Laplace transform]] of matrix exponentials amounts to the [[resolvent formalism|resolvent]],
<math display="block">\int_0^\infty e^{-ts}e^{tX}\,dt = (sI - X)^{-1}</math>
for all sufficiently large positive values of {{mvar|s}}.