Editing Matrix exponential (section)

== The exponential map ==
The exponential of a matrix is always an [[invertible matrix]]. The inverse matrix of {{math|''e''<sup>''X''</sup>}} is given by {{math|''e''<sup>−''X''</sup>}}. This is analogous to the fact that the exponential of a complex number is always nonzero. The matrix exponential then gives us a map
<math display="block">\exp \colon M_n(\Complex) \to \mathrm{GL}(n, \Complex)</math>
from the space of all ''n'' × ''n'' matrices to the [[general linear group]] of degree {{mvar|n}}, i.e. the [[group (mathematics)|group]] of all ''n'' × ''n'' invertible matrices. In fact, this map is [[surjective]] which means that every invertible matrix can be written as the exponential of some other matrix<ref>{{harvnb|Hall|2015}} Exercises 2.9 and 2.10</ref> (for this, it is essential to consider the field '''C''' of complex numbers and not '''R''').

For any two matrices {{mvar|X}} and {{mvar|Y}},
<math display="block">\left\| e^{X+Y} - e^X\right\| \le \|Y\| e^{\|X\|} e^{\|Y\|}, </math>

where {{math|‖ · ‖}} denotes an arbitrary [[matrix norm]]. It follows that the exponential map is [[continuity (mathematics)|continuous]] and [[Lipschitz continuous]] on [[compact set|compact]] subsets of {{math|''M''<sub>''n''</sub>('''C''')}}.

The map
<math display="block">t \mapsto e^{tX}, \qquad t \in \R</math>
defines a [[Smooth function#Smoothness|smooth]] curve in the general linear group which passes through the identity element at {{math|1=''t'' = 0}}.

In fact, this gives a [[one-parameter subgroup]] of the general linear group since
<math display="block">e^{tX}e^{sX} = e^{(t + s)X}.</math>

The derivative of this curve (or [[tangent vector]]) at a point ''t'' is given by
{{NumBlk||<math display="block">\frac{d}{dt}e^{tX} = Xe^{tX} = e^{tX}X.</math>|{{EquationRef|1}}}}
The derivative at {{math|1=''t'' = 0}} is just the matrix ''X'', which is to say that ''X'' generates this one-parameter subgroup.

More generally,<ref>{{cite journal|doi=10.1063/1.1705306 | author = R. M. Wilcox | title=Exponential Operators and Parameter Differentiation in Quantum Physics | journal=Journal of Mathematical Physics | volume=8 | pages=962–982 | year=1967|issue=4| bibcode = 1967JMP.....8..962W }}</ref> for a generic {{mvar|t}}-dependent exponent, {{math|''X''(''t'')}},
{{Equation box 1
|indent =
|equation = <math>\frac{d}{dt}e^{X(t)} = \int_0^1 e^{\alpha X(t)} \frac{dX(t)}{dt} e^{(1 - \alpha) X(t)}\,d\alpha ~.  </math>
|cellpadding=
|border
|border colour = #0073CF
|bgcolor=#000000
}}

Taking the above expression {{math|''e''<sup>''X''(''t'')</sup>}} outside the integral sign and expanding the integrand with the help of the [[Baker–Campbell–Hausdorff formula|Hadamard lemma]] one can obtain the following useful expression for the derivative of the matrix exponent,<ref>{{harvnb|Hall|2015}} Theorem 5.4</ref>
<math display="block">e^{-X(t)}\left(\frac{d}{dt}e^{X(t)}\right) = \frac{d}{dt}X(t) - \frac{1}{2!} \left[X(t), \frac{d}{dt}X(t)\right] + \frac{1}{3!} \left[X(t), \left[X(t), \frac{d}{dt}X(t)\right]\right] - \cdots </math>

The coefficients in the expression above are different from what appears in the exponential. For a closed form, see [[derivative of the exponential map]].

=== Directional derivatives when restricted to Hermitian matrices ===

Let <math>X</math> be a <math>n \times n</math> Hermitian matrix with distinct eigenvalues. Let <math>X = E \textrm{diag}(\Lambda) E^*</math> be its eigen-decomposition where <math>E</math> is a unitary matrix whose columns are the eigenvectors of <math>X</math>, <math>E^*</math> is its conjugate transpose, and <math>\Lambda = \left(\lambda_1, \ldots, \lambda_n\right)</math> the vector of corresponding eigenvalues. Then, for any <math>n \times n</math> Hermitian matrix <math>V</math>, the [[directional derivative]] of <math>\exp: X \to e^X</math> at <math>X</math> in the direction <math>V</math> is
<ref name="lewis">{{cite journal |
first1=Adrian S. |
last1=Lewis |
first2=Hristo S. |
last2=Sendov |
title=Twice differentiable spectral functions |
journal=SIAM Journal on Matrix Analysis and Applications |
volume=23 |
issue=2 |
pages=368–386 |
date=2001 |
doi=10.1137/S089547980036838X |
url=https://people.orie.cornell.edu/aslewis/publications/01-twice.pdf
}} See Theorem 3.3.</ref>
<ref name="deledalle">{{cite journal |
first1=Charles-Alban |
last1=Deledalle |
first2=Loïc |
last2=Denis |
first3=Florence |
last3=Tupin |
title=Speckle reduction in matrix-log domain for synthetic aperture radar imaging |
journal=Journal of Mathematical Imaging and Vision |
date=2022 |
volume=64 |
issue=3 |
pages=298–320 |
doi=10.1007/s10851-022-01067-1 | doi-access=free
|
bibcode=2022JMIV...64..298D }} See Propositions 1 and 2.
</ref>
<math display="block">
D \exp (X) [V]
\triangleq
\lim_{\epsilon \to 0} 
\frac{1}{\epsilon}
\left(\displaystyle
e^{X + \epsilon V}
-
e^{X}
\right)
=
E(G \odot \bar{V}) E^*
</math>
where <math>\bar{V} = E^* V E</math>, the operator <math>\odot</math> denotes the Hadamard product, and, for all <math>1 \leq i, j \leq n</math>, the matrix <math>G</math> is defined as
<math display="block">
G_{i, j} = \left\{\begin{align}
& \frac{e^{\lambda_i} - e^{\lambda_j}}{\lambda_i - \lambda_j} & \text{ if } i \neq j,\\
& e^{\lambda_i} & \text{ otherwise}.\\
\end{align}\right.
</math>
In addition, for any <math>n \times n</math> Hermitian matrix <math>U</math>, the second directional derivative in directions  <math>U</math> and <math>V</math> is<ref name="deledalle"/>
<math display="block">
D^2 \exp (X) [U, V]
\triangleq
\lim_{\epsilon_u \to 0} 
\lim_{\epsilon_v \to 0} 
\frac{1}{4 \epsilon_u \epsilon_v}
\left(\displaystyle
e^{X + \epsilon_u U + \epsilon_v V}
-
e^{X - \epsilon_u U + \epsilon_v V}
-
e^{X + \epsilon_u U - \epsilon_v V}
+
e^{X - \epsilon_u U - \epsilon_v V}
\right)
=
E F(U, V) E^*
</math>
where the matrix-valued function <math>F</math> is defined, for all <math>1 \leq i, j \leq n</math>, as
<math display="block">
F(U, V)_{i,j} = \sum_{k=1}^n \phi_{i,j,k}(\bar{U}_{ik}\bar{V}_{jk}^* + \bar{V}_{ik}\bar{U}_{jk}^*)
</math>
with
<math display="block">
\phi_{i,j,k} = \left\{\begin{align}
& \frac{G_{ik} - G_{jk}}{\lambda_i - \lambda_j} & \text{ if } i \ne j,\\
& \frac{G_{ii} - G_{ik}}{\lambda_i - \lambda_k} & \text{ if } i = j \text{ and } k \ne i,\\
& \frac{G_{ii}}{2} & \text{ if } i = j = k.\\
\end{align}\right.
</math>