This question is crossposted from Math Stackexchange here. I crosspost without much edits as I think this is the best way to phrase the question and because I received no feedback on the original post in Stackexchange.
Let $ \mathcal H = \mathbb{C}^d$ be a finite dimensional Hilbert space and let $X, Y \in \text{Pd}(\mathcal H)$ be two Hermitian positive definite matrices. Let $A = XY$, let $A = UDV^*$ be its SVD, and let
$$ A = PW = (UDU^*)(UV^*), \\ A = WQ = (UV^*)(VDV^*). $$ be the left and right polar decompositions of $A$. Also note that $ Q = |A| = \sqrt{A^*A} = VDV^*$ and $P = |A^*| = \sqrt{AA^*} = UDU^*$. We have $U,V,W$ to be unitary matrices.
If $X$ and $Y$ commute, then we have $A$ to be normal or $[A,A^*] = 0$. Then the SVD of $A$ reduces to its eigendecomposition with $U=V$ and thereby $A = UDU^*$. Moreover, the unitary $W = UV^*$ from the Polar decomposition would reduce to the identity matrix $\mathbb{I}$. Succinctly, $$[X,Y] = 0 \quad \Rightarrow \quad [A,A^*] = 0 \quad \Rightarrow \quad W = UV^* = \mathbb I. $$ I think it should also hold that $$[X,Y] \to 0 \quad \Rightarrow \quad [A,A^*] \to 0 \quad \Rightarrow W = UV^* \to \mathbb I. $$
I am interested in relating $[X,Y]$ to $W$ or to $H$ where $W = \exp (iH)$. Clearly, if $[X,Y] = 0$, then $H = 0$ as $W = \mathbb I$ . I think in the case where $\| [X,Y]\|$ is small for some matrix norm $\| \cdot \|$, or where $X$ and $Y$ nearly commute, would $W$ be nearly identity or would $H$ be nearly $0$?
Specifically, I want to relate $\left\|[X,Y] \right\|$ to $\|H\|$ or what is the error in terms of $[X,Y]$ in the approximation $W \approx \mathbb I$ when $[X,Y] \neq 0$? This is the same as ignoring every term in the series expansion of $W = \mathbb I + (iH) + (iH)^2/2 + \cdots$ except the very first.
After running some simulations, I saw that for $2 \times 2$ matrices, we have $H \propto [X,Y] $, though the constant of proportionality varied for different pairs of $X,Y$. However, this relation is not observed for matrices of higher dimensions.
