All answers to questions like this dodge the question by saying it's a postulate of Matrix Mechanics, so let me rephrase it. Instead of how to derive the CCR, how does it follow from Heisenberg's matrices: $$\hat X=\{X_{mn}e^{i\omega_{mn}t}\}\quad\hat P=\{P_{mn}e^{i\omega_{mn}t}\}$$ Where $\omega_{mn}=\frac{E_m-E_n}{\hbar}$, such that $$\hat X(t)=\left(\begin{matrix} X_{11}&X_{12}e^{i\omega_{12}t}&X_{13}e^{i\omega_{13}t}&\cdots\\ X_{21}e^{i\omega_{21}t}&X_{22}&X_{23}e^{i\omega_{23}t}&\cdots\\ X_{31}e^{i\omega_{31}t}&X_{32}e^{i\omega_{32}t}&X_{33}\cdots\\ \vdots&\vdots&\vdots&\ddots \end{matrix}\right)$$ $$\hat P(t)=\left(\begin{matrix} P_{11}&P_{12}e^{i\omega_{12}t}&P_{13}e^{i\omega_{13}t}&\cdots\\ P_{21}e^{i\omega_{21}t}&P_{22}&P_{23}e^{i\omega_{23}t}&\cdots\\ P_{31}e^{i\omega_{31}t}&P_{32}e^{i\omega_{32}t}&P_{33}\cdots\\ \vdots&\vdots&\vdots&\ddots \end{matrix}\right)$$ And from the Old quantum condition: $$\oint pdx=nh$$ I know Heisenberg came up with it through analogies between the matrices and the classical observables, but I'd like to know how he did it.