----- > [!proposition] Proposition. ([[zero-trace exponentiates to special-linearity under suitable conditions, and vice versa]]) > Denote by $\text{SL}(n, \mathbb{R})=\{ A \in \text{GL}_{n}(\mathbb{R}): \det A = 1 \}$ the [[special linear group]] over $\mathbb{R}$. Then, for 'small enough' $A \in \text{SL}(n,\mathbb{R})$, such that the [[matrix logarithm]] applied to $A$ is [[well-defined]], put $B=\log A$, so that $A=e^{B}$. Then $\text{tr }B=0$. > Conversely, given a 'small enough' matrix $B \in \mathbb{R}^{n \times n}$ satisfying $\text{tr }B=0$, one has $e^{B}=A \in \text{SL}(n , \mathbb{R})$. ^proposition > [!note] Note. > This result has an analogue for orthogonal matrices; see [[skew-symmetry exponentiates to orthogonality and vice versa, under suitable conditions]]. ^note > [!proof]+ Proof. ([[zero-trace exponentiates to special-linearity under suitable conditions, and vice versa]]) > Use [[determinant of matrix exponential]]. We have $1=\det A = \det e^{B}=e^{\text{tr }B}.$ For any $x \in \mathbb{R}$ one has $e^{x}=1 \iff x=0$, and therefore $\text{tr }B=0$. > Conversely, given $B \in \mathbb{R}^{n \times n}$ such that $\text{tr }B=0$, compute $\det e ^{B}=e ^{0}=1$ and so $e^{B}=A \in \text{SL}_{n}(\mathbb{R})$ as was required. ----- #### ---- #### 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 > ```