I am trying to find a certain proof of polar decomposition of complex matrices which I think should exist more generally for a certain class of Lie groups. Recall that the polar decomposition of a nonsingular complex matrix $A \in \text{GL}_n(\mathbb{C})$ is an expression $A = UP$, where $U \in \text{GL}_n(\mathbb{C})$ is unitary and $P \in \text{GL}_n(\mathbb{C})$ is positive definite.

My attempt arises with a well known analogy between four classes of matrices and four sets of complex numbers:

\begin{array}{|c|c|} \hline \text{Subset of }\text{Mat}_{n \times n } (\mathbb{C}) & \text{Corresponding set in } \mathbb{C} \text{ } \\ \hline \text{Hermitian matrices} & \text{Real numbers}\\ \hline \text{Skew-Hermitian matrices} & \text{Imaginary numbers} \\ \hline \text{Hermitian positive definite matrices} & \text{Positive real numbers} \\ \hline \text{Unitary matrices} & \text{The unit circle in } \mathbb{C}\\ \hline \end{array}

The analogy is justified through the observation that each set of matrices in the left column is precisely the set of unitarily diagonalizable matrices with eigenvalues in the corresponding set on the right. Moreover, the exponential map of matrices sends hermitian matrices to positive definite matrices and skew-hermitian to unitary matrices, a further similarity.

In showing polar decomposition, we take from the case for $\mathbb{C}$; $\text{exp} : \mathbb{C} \rightarrow \mathbb{C}^*$ sends the reals to the positive reals, and imaginary numbers to the unit circle. Taking $\text{exp}$ of $z = \text{Re}(z) + i \ \text{Im}(z)$ gives us the polar decomposition $\text{exp}(z) = e^{\text{Re}(z)} e^{i\ \text{Im}(z)}$ into a positive real number and a complex number on the unit circle. This analogy is presumably where polar decomposition gets its name.

Put $\mathfrak{g} = \text{Mat}_{n \times n } (\mathbb{C})$, the Lie algebra of $G = \text{GL}_n (\mathbb{C})$. I would like to use the decomposition $\mathfrak{g} = i \mathfrak{h} \oplus \mathfrak{h}$, where $\mathfrak{h}$ is the sub-$\mathbb{C}$-vector space of hermitian complex matrices and $i \mathfrak{h}$ is the sub-Lie algebra of skew-Hermitian matrices to achieve a decomposition $A = UP$ of any $A \in \text{GL}_n (\mathbb{C})$ into a unitary matrix $U$ and a positive definite matrix $P$, making use of the exponential map. However, there is an obvious obstruction: $\text{exp}(A + B)$ is not $\text{exp}(A) \text{exp}(B)$ necessarily. So the question is, how might we work around this problem?

Now, the subgroup $U \subset G$ of unitary matrices acts on the sub-$\mathbb{C}$-vector space $P \subset G$ of hermitian positive definite matrices by $(U, P) \mapsto U^* P U$, so this theorem will show that $G$ has a semi-direct product decomposition $G = U \times P$ with product $(U, P)(U', P') \mapsto (UU' , U'^* P U' P)$.

It is possibly useful to note that the map $i \mathfrak{h} \rightarrow \text{End}_{\mathbb{C}}(\mathfrak{h})$ sending $s$ to $\text{ad}_s$, where $\text{ad}_s (h) = sh - hs$ is a Lie-algebra representation.

Finite Dimensional Vector Spaces. $\endgroup$ – Jim Humphreys Oct 13 '18 at 14:49