Editing Matrix exponential (section)

== Properties ==
=== 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}}.

=== Linear differential equation systems ===

{{Main|Matrix differential equation}}

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
<math display="block"> \frac{d}{dt} y(t) = Ay(t), \quad y(0) = y_0, </math>
where {{mvar|A}} is a constant matrix and ''y'' is a column vector, is given by
<math display="block"> y(t) = e^{At} y_0. </math>

The matrix exponential can also be used to solve the inhomogeneous equation
<math display="block"> \frac{d}{dt} y(t) = Ay(t) + z(t), \quad y(0) = y_0. </math>
See the section on [[#Applications|applications]] below for examples.

There is no closed-form solution for differential equations of the form
<math display="block"> \frac{d}{dt} y(t) = A(t) \, y(t), \quad y(0) = y_0, </math>
where {{mvar|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:<ref>{{harvnb|Hall|2015}} Theorem 2.12</ref>
{{Equation box 1
|indent =
|equation = <math>\det\left(e^A\right) = e^{\operatorname{tr}(A)}~.</math>
|cellpadding = 6
|border
|border colour = #0073CF
|bgcolor = #000000
}}

In addition to providing a computational tool, this formula demonstrates that a matrix exponential is always an [[invertible matrix]]. This follows from the fact that the right hand side of the above equation is always non-zero, and so {{math|det(''e<sup>A</sup>'') ≠ 0}}, which implies that {{math|''e<sup>A</sup>''}} must be invertible.

In the real-valued case, the formula also exhibits the map
<math display="block">\exp \colon M_n(\R) \to \mathrm{GL}(n, \R)</math>
to not be [[surjective function|surjective]], in contrast to the complex case mentioned earlier. This follows from the fact that, for real-valued matrices, the right-hand side of the formula is always positive, while there exist invertible matrices with a negative determinant.

=== Real symmetric matrices ===

The matrix exponential of a real symmetric matrix is positive definite. Let <math>S</math> be an {{math|''n'' × ''n''}} real symmetric matrix and <math>x \in \R^n</math> a column vector. Using the elementary properties of the matrix exponential and of symmetric matrices, we have:

<math display="block">x^Te^Sx=x^Te^{S/2}e^{S/2}x=x^T(e^{S/2})^Te^{S/2}x =(e^{S/2}x)^Te^{S/2}x=\lVert e^{S/2}x\rVert^2\geq 0.</math>

Since <math>e^{S/2}</math> is invertible, the equality only holds for <math>x=0</math>, and we have <math>x^Te^Sx > 0</math> for all non-zero <math>x</math>. Hence <math>e^S</math> is positive definite.