----- > [!proposition] Proposition. ([[skew-symmetry exponentiates to orthogonality and vice versa, under suitable conditions]]) > Let $A$ belong to the [[general orthogonal group]] $\text{O}(n)$ with $|A-I|=n \max_{i,j}(a_{ij}-i_{ij})<1$. [[matrix logarithm|We know that]] this ensures $B=\log A$ is [[well-defined]], [[matrix exponential|and]] $\exp B=A$ We can also choose $0<\varepsilon<1$ such that if $|A-I| < \varepsilon$ then $|B| < \log 2$. Then we have that $B$ is [[skew-symmetric matrix|skew-symmetric]]. > Conversely, given a [[skew-symmetric matrix]] $B$ satisfying $|B|<\log 2$, one has that $A=\exp B$ is [[orthogonal matrix|orthogonal]]. ^proposition > [!proof]- Proof. ([[skew-symmetry exponentiates to orthogonality and vice versa, under suitable conditions]]) > The stipulated conditions guarantee that $\exp(\log(B))=B$ and $\log(\exp(A))=A$. We will hereon liberally employ properties of the [[matrix exponential]]. We have $e^{B}e^{B^{\top}}=A A^{\top}=I,$ so that $e^{B}=(e^{B^{\top}})^{-1}=e^{-B^{\top}}$ now in taking the [[matrix logarithm]] we see that $B$ is [[skew-symmetric matrix]] (here we needed to use that $|B| < \log 2$). > Conversely, suppose that [[matrix]] $B$ is [[skew-symmetric matrix|skew-symmetric]] with $|B|<\log 2$. Then $(e^{B})^{\top}= e^{(B^{\top})}= e^{-B}=(e^{B})^{-1}$ and $e^{B}=A$ is [[orthogonal matrix|is orthogonal]]. ----- #### ---- #### 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 > ```