Editing Matrix exponential (section)

{{Short description|Matrix operation generalizing exponentiation of scalar numbers}}
{{Use American English|date = January 2019}}
In [[mathematics]], the '''matrix exponential''' is a [[matrix function]] on [[square matrix|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 (Lie theory)|exponential map]] between a matrix [[Lie algebra]] and the corresponding [[Lie group]].

Let {{mvar|X}}  be an {{math|''n'' × ''n''}} [[real number|real]] or [[complex number|complex]] [[matrix (mathematics)|matrix]]. The exponential of {{mvar|X}}, denoted by {{math|''e''<sup>''X''</sup>}} or {{math|exp(''X'')}}, is the {{math|''n'' × ''n''}} matrix given by the [[power series]]

<math display="block">e^X = \sum_{k=0}^\infty \frac{1}{k!} X^k</math>

where <math>X^0</math> is defined to be the identity matrix <math>I</math> with the same dimensions as <math>X</math>, and {{tmath|1=X^k = X X^{{mset|k-1}}}}.<ref>{{harvnb|Hall|2015}} Equation 2.1</ref> The series always converges, so the exponential of {{mvar|X}} is well-defined.

Equivalently,
<math display="block">e^X = \lim_{k \rightarrow \infty} \left(I + \frac{X}{k} \right)^k</math>

for integer-valued {{mvar|k}}, where {{mvar|I}} is the {{math|''n'' × ''n''}} [[identity matrix]]. 

Equivalently, given by the solution to the differential equation

<math display="block">\frac d {dt} e^{X t} = X e^{X t}, \quad e^{X 0} = I</math>

When {{mvar|X}} is an {{math|''n'' × ''n''}} [[diagonal matrix]] then {{math|exp(''X'')}} will be an {{math|''n'' × ''n''}} diagonal matrix with each diagonal element equal to the ordinary [[Exponential function|exponential]] applied to the corresponding diagonal element of {{mvar|X}}.