----- > [!proposition] Proposition. ([[matrix exponential and transpose commute]]) > Let $A \in \mathbb{R}^{n \times n}$. We have $\exp(A^{\top})=\exp(A)^{\top},$ > where $\exp$ denotes the [[matrix exponential]]. ^proposition > [!proof]- Proof. ([[matrix exponential and transpose commute]]) Write $\begin{align} \exp(A^{\top}) & = I_{n} + A^{\top} + \frac{{(A^{\top})}^{2}}{2} + \frac{({A^{\top}})^{3}}{3!} + \dots \\ \end{align}$ which equals $\exp(A^{})^{\top}$ provided that it is okay to distribute the transposition operator $\top$. Indeed, this is okay, because transpose commutes with limits. ----- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```