---- > [!definition] Definition. ([[matrix logarithm]]) > For a matrix $M \in \mathbb{R}^{n \times n}$, formally define $\log(I + M)= \sum_{n=0}^{\infty} (-1)^{n} \frac{M^{n+1}}{(n+1)!}=M - \frac{M^{2}}{2} + \frac{M^{3}}{3} - \dots .$ > This [[series]] [[series|converges absolutely]] for $|M|:=n \max_{i,j \in [n]} (|m_{ij}|) < 1$, and [[uniform convergence|uniformly]] on any closed subset $\{ |M| \leq \varepsilon \}$, $\varepsilon < 1$, to a [[continuously differentiable|smooth]] function of $A$. > One has $\exp(\log(A))=A$ if $|A-I| < 1$, and $\log(\exp(A))=A$ when $|A|<\log 2$. ^definition ---- #### ---- #### 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 > ```