A _complex matrix exponential_ is a [matrix exponential](Matrix%20exponentials.md) of the form $f(X)=e^{itX}$ ^cc2e9f where the exponent is written with the complex number $i$ and $t\in\mathbb{R}.$ # Power series representation of complex matrix exponentials The power series representation of a [complex matrix exponential](Complex%20matrix%20exponentials.md) follows from plugging in $i$ into the general [power series representation](Matrix%20exponentials.md#Power%20series%20representation%20of%20matrix%20exponentials) for a matrix exponential. Thus we may write $e^{itX} = \sum_{n=0}^{\infty} \frac{(itX)^n}{n!}.$ # Properties of complex matrix exponentials 1) The [adjoint](Adjoint.md) of $e^{itX}$ is $e^{-itX^\dagger}$. 2) If $X$ is a [Hermitian operator](Hermitian%20operators.md), then $e^{itX}$ is [unitary](Unitary%20operators.md). 3) The exponent $iX$ is an element of a [Lie algebra](Lie%20algebras.md) of [[U(n)]]. Property 1. can be shown by writing out the [matrix exponential](Matrix%20exponentials.md) in its [power series representation](Matrix%20exponentials.md#Power%20series%20representation%20of%20matrix%20exponentials) and then taking the adjoint of the expansion. Property 2. is true because if $X^\dagger= X$ then $(e^{itX})^\dagger = e^{-itX^\dagger} = e^{-itX}$ and $e^{-itX}e^{itX} = e^0 = I.$ Property 3. follows directly from property 2 given the definition of Lie Algebra. Property 3. may be described in terms of the [role of complex matrix exponentials in Lie groups and algebras.](Complex%20matrix%20exponentials.md#Role%20of%20complex%20matrix%20exponentials%20in%20Lie%20groups%20and%20algebras) # Complex matrix exponentials as limits of sequences It follows from the construction of the [matrix exponential](Matrix%20exponentials.md) [as a limit of a sequence](Matrix%20exponentials.md#matrix%20exponentials%20as%20limits%20of%20sequences) that $e^{itX} = \lim_{t\rightarrow 0}\big(1+itX\big)^{1/t}$ # Role of complex matrix exponentials in Lie groups and algebras Note the central role of [matrix exponentials in Lie groups and Lie algebras:](Matrix%20exponentials.md#Role%20of%20matrix%20exponentials%20in%20Lie%20groups%20and%20Lie%20algebras)  %%rewrite this page as a page on _complex matrix operators,_ with _complex matrix exponentials as a subsection_%% #MathematicalFoundations/Algebra/AbstractAlgebra/GroupTheory/Lie/LieGroups #MathematicalFoundations/Algebra/AbstractAlgebra/LinearAlgebra/Operators/Matrices