Let $A\in\mathbb{M}_{n\times n}(\mathbb{C})$. $A$ is non-singular. Then $A=UR$ where $U$ is a unitary matrix, and $R$ is a positive-semidefinite Hermitian matrix. I understand the intuition that this is analogous to $U$ encoding some $\theta$ and $R$ encoding the modulus/radius $r$, taking the complex plane in polar coordinates. I have seen it confidently written that $|A|=|U|\cdot|R|=\exp(i\theta)\cdot r$. Intuitively, by the definition above, this should make sense, since any unit square/cube/etc. would be rotated and scaled under $A$, and this determinant agrees with that. However, being more formal, I wonder how a person might prove this, or find $\theta.$
$|\exp(A)|=\exp(tr(A))$, and also any unitary $U=\exp(iH)$, where $H$ is Hermitian. Therefore $|U|=|\exp(iH)|=\exp(tr(iH))=\exp(i\cdot tr(H))$. If $|U|$ is to be $\exp(i\theta)$, then the mystery $H$ must have a trace equal to $\theta$. Is this correct?
As a complete side-question from the main focus of this post, is the finding of $H$ a messy problem almost always left to computers - can the trace=$\theta$ idea be leveraged for this? And when Wikipedia mentioned $U=\exp(iH)$, did they mention it out of trivia, or because of some utility, some application of this fact?