Polar cartan decomposition of sl n r

Put $P:=\sqrt{g^{\mathsf{T}}g}$, the non-negative square root of the self-adjoint positive definite matrix $g^{\mathsf{T}}g$. By [L3] the matrix $P$ is self-adjoint with an orthonormal eigenbasis and positive eigenvalues ${\lambda }_1,\dots ,{\lambda }_n>0$, because $g$ is invertible and $⟨g^{\mathsf{T}}gv,v⟩=\parallel gv{\parallel }^2>0$ for $v\mathrm{\ne }0$.

The group $\mathrm{SO}(n)$ is path connected. For a unit vector $v$, a rotation of the plane spanned by $v,e_1$ sends $v$ to $e_1$ and is joined to the identity by varying its angle. If $v=-e_1$, use a rotation through $\pi$ in the $(e_1,e_2)$ plane; if $v=e_1$, use the identity. For $R\in \mathrm{SO}(n)$ apply this to its first column; after this rotation the matrix is $\mathrm{diag}(1,R^{\mathrm{\prime }})$ with $R^{\mathrm{\prime }}\in \mathrm{SO}(n-1)$. Induction, ending with $\mathrm{SO}(1)=\{1\}$, expresses every $R$ as a product of rotations, each with a path to the identity.

The determinant of $P$ is $1$: $\det (P{)}^2=\det (P^2)=\det (g^{\mathsf{T}}g)=\det (g^{\mathsf{T}})\det (g)=(\det g{)}^2=1$ by [L6] and $\det g=1$, while $\det P={\lambda }_1\cdots {\lambda }_n>0$ by step 1.1, so $\det P=1$.

Define $X:=\log P$ by the spectral decomposition $P=Q\mathrm{diag}({\lambda }_1,\dots ,{\lambda }_n)Q^{\mathsf{T}}$ with $Q$ orthogonal and $X:=Q\mathrm{diag}(\log {\lambda }_1,\dots ,\log {\lambda }_n)Q^{\mathsf{T}}$. Then $X$ is self-adjoint, and $X^k=Q\mathrm{diag}((\log {\lambda }_1{)}^k,\dots ,(\log {\lambda }_n{)}^k)Q^{\mathsf{T}}$ for every $k\ge 0$, so the matrix exponential of [L4] gives $\exp X=Q\mathrm{diag}({\lambda }_1,\dots ,{\lambda }_n)Q^{\mathsf{T}}=P$.

Define $k:=g{P}^{-1}$, which is well defined because $P$ is invertible with positive eigenvalues. Then $k^{\mathsf{T}}k=P^{-1}{g}^{\mathsf{T}}g{P}^{-1}=P^{-1}{P}^2{P}^{-1}=I$ and $\det k=\det g\det P^{-1}=1$ by [L6], so $k\in \mathrm{SO}(n)$ by [L1].

The trace of $X$ vanishes: $\mathrm{tr}X={\sum }_{i=1}^n\log {\lambda }_i=\log ({\lambda }_1\cdots {\lambda }_n)=\log \det P=0$ by step 2.1, so $X\in {\mathfrak{p}}_0$ by [L7].

Existence: steps 3.1, 2.2 and 3.2 give $g=kP=k\exp X$ with $k\in \mathrm{SO}(n)$ and $X\in {\mathfrak{p}}_0$.

Define $\mathrm{\Theta }(g)=(g^{\mathsf{T}}{)}^{-1}$ on $G={\mathrm{SL}}_n(\mathbb{R})$. It is a smooth involutive automorphism, differentiates to $-X^{\mathsf{T}}$, and has fixed group $\mathrm{SO}(n)$. To compute the center, a central matrix commutes with $I+t{E}_{ij}\in G$ for every $i\mathrm{\ne }j$ and real $t$, hence with every $E_{ij}$. Comparing entries of $z{E}_{ij}=E_{ij}z$ forces all off-diagonal entries of $z$ to vanish and its diagonal entries to be equal. Thus $z=\lambda I$ with real ${\lambda }^n=1$, so the center consists of $I$ and, only when $n$ is even, $-I$. It is finite and fixed pointwise by $\mathrm{\Theta }$. Semisimplicity and the Cartan differential are [L7]. Finally, $G$ is path connected without assuming the global theorem: by step 4.1, $g=k\exp X$, and $t\mapsto k\exp (tX)$ joins $k$ to $g$ inside $G$, since diagonalization gives $\det \exp (tX)=\exp (t\mathrm{tr}X)=1$. Step 1.2 joins $I$ to $k$. Every hypothesis of [L5] is therefore established.

Uniqueness: suppose $g=k\exp X=k^{\mathrm{\prime }}\exp X^{\mathrm{\prime }}$ with $k,k^{\mathrm{\prime }}\in \mathrm{SO}(n)$ and $X,X^{\mathrm{\prime }}\in {\mathfrak{p}}_0$. Both $\exp X$ and $\exp X^{\mathrm{\prime }}$ are self-adjoint positive definite, since the eigenvalues of $X$ and $X^{\mathrm{\prime }}$ are real by self-adjointness and exponentiate to positive numbers, and both $k,k^{\mathrm{\prime }}$ are orthogonal. By the uniqueness clause of [L2] for the invertible element $g$ the non-negative factor is unique, so $\exp X=\exp X^{\mathrm{\prime }}=P$ and $k=k^{\mathrm{\prime }}$; then $X=X^{\mathrm{\prime }}$ because the self-adjoint logarithm is unique: both $X$ and $X^{\mathrm{\prime }}$ commute with $P=\exp X=\exp X^{\mathrm{\prime }}$, hence preserve each eigenspace of $P$, and on the eigenspace for the eigenvalue $\lambda >0$ the equality $e^X=e^{X^{\mathrm{\prime }}}=\lambda I$ forces every eigenvalue of the self-adjoint operator $X{\mathrm{\mid }}_{E_{\lambda }}$ to lie in $\log \lambda +2\pi i\mathbb{Z}$, hence to equal the real number $\log \lambda$.

Consequently the map $\mathrm{SO}(n)\times {\mathfrak{p}}_0\to {\mathrm{SL}}_n(\mathbb{R})$, $(k,X)\mapsto k\exp X$, is a bijection by steps 4.1 and 5.2 and a diffeomorphism by [L5] applied as in step 5.1; its inverse is $g\mapsto (g{P}^{-1},\log P)$ with $P=\sqrt{g^{\mathsf{T}}g}$.

The argument includes repeated eigenvalues, since the logarithm is scalar on each positive eigenspace. The zero logarithm gives the orthogonal elements of $G$. Although $n\ge 2$ is the stated scope, at $n=1$ both factors and the group are singletons. More generally the orthogonal polar factor is special orthogonal whenever $\det g>0$, since $\det P=\mathrm{\mid }\det g\mathrm{\mid }$; the stronger condition $\det g=1$ additionally makes $\mathrm{tr}\log P=0$. AC covers the cited Lie-group and global Cartan interfaces; no arbitrary eigenbasis selection over an indexed family is needed in the finite matrix calculations.