TITLE: What's the Lie algebra associated to positive-defined Hermitian matrices?
QUESTION [7 upvotes]: I'm studying polar decomposition $GL(n,\Bbb C)\cong U(n)\times Herm^+(n)$, where $Herm^+(n)$ are positive-defined Hermitian matrices. I was trying to understand what happens to the associated Lie algebras, and I was wondering what $Herm^+(n)$ Lie algebra would look like.
What I tried:
I know that considering $F:SL(n,\Bbb R)\to SL(n,\Bbb R):A\mapsto A^\top A$, since $SO(n)=F^{-1}(I_n)$, we can prove that $\mathfrak{so}(n)$ is the space of skew-symmetric matrices by calculating $$\mathfrak{so}(n)=\ker (dF_{I_n})=\left\{A\in\Bbb R^{n\times n} : \left. \frac d{dt}\right\vert_{t=0}F(I_n+tA)=0\right\}.$$
I wondered if I could do something similar for $Herm(n)$, defining $G:\Bbb C^{n\times n}\to\Bbb C^{n\times n}:A\mapsto A^*-A$ and $Herm(n)=G^{-1}(O_n)$, but I then realised that being $Herm(n)$ a linear subspace of $\Bbb C^{n\times n}$ I would just find $Herm(n)$ again. Is this a correct answer? Is $Herm(n)$ itself its own associated Lie algebra (with the commutator as Lie bracket)?
And what about positive-defined ones? What useful characterization may I take advantage of?
Thanks in advance!

REPLY [9 votes]: Hermitian matrices (positive-definite or otherwise) aren't closed under multiplication so they aren't a Lie group and don't have a Lie algebra. They also aren't closed under commutator bracket so they aren't themselves a Lie algebra either. However, they are $i$ times the skew-Hermitian matrices $\mathfrak{u}(n)$, which do form a Lie algebra (this requires that we ignore positive-definiteness), and we do have
$$\mathfrak{gl}_n(\mathbb{C}) \cong \mathfrak{u}(n) \oplus i \mathfrak{u}(n).$$
In other words, every matrix $M$ is uniquely the sum of a skew-Hermitian matrix $\frac{M - M^{\dagger}}{2}$ and a Hermitian matrix $\frac{M + M^{\dagger}}{2}$. This is the Lie algebra incarnation of polar decomposition; despite not being a Lie algebra, there is still an exponential map from Hermitian matrices to positive-definite Hermitian matrices.