EDB — 1N0

Exercises

1. [1N0]Prerequisites:[1T1]↺↻.Difficulty:**.In the general case (even when we do not know if \(A,B\) commute), we can express \(\exp (A+sB)\) using a power series. Define

   \[ C(t)=\exp (-tA)B \exp (tA) \]

   and (recursively) set \(Q_ 0={\mathbb {I}}\) (the identity matrix) and then

   \[ Q_{n+1}(t)=∫_ 0^ t C(𝜏) Q_ n(𝜏)\, {\mathbb {d}}𝜏 \]

   then

   \begin{equation} \exp (-A)\exp (A+sB)=∑_{n=0}^∞ s^ n Q_ n(1)~ ~ ;\label{eq:exp_ A+sB} \end{equation}

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   this series converges for every \(s\).

   In particular, the directional derivative of \(\exp \) at the point \(A\) in the direction \(B\) is

   \[ \frac{d\hskip5.5pt}{d{s}} \exp (A+sB)|_{s=0}=\exp (A) Q_ 1(1) = ∫_ 0^ 1 \exp ((1-𝜏) A)B\exp (𝜏 A) \, {\mathbb {d}}𝜏~ ~ . \]

   ( Hint: Use the exercise [1T1]↺↻ with \(Y(t,s) = \exp (-tA)\exp (t(A+sB))\) and then set \(t=1\). )

   Solution 1

   [1N1]↺↻