Editing Matrix exponential (section)

=== 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>