In this Numberphile video they discuss Dirac changing to matrix equations to get something like a "square root" of $t^2 + x^2 + y^2 + x^2$. In particular they discuss the following multivariate polynomial which is transformed to a matrix equation to get a kind of square root for this polynomial: $$(xy - uv) \longrightarrow \begin{bmatrix} 0 & 0 & x & u \\ 0 & 0 & v & y \\ y & -u & 0 & 0 \\ -v & x & 0 & 0 \\ \end{bmatrix}^2 = (xy-uv)I$$
They then mention this work:
https://www.jstor.org/stable/1999875?seq=1
Eisenbud, David. "Homological Algebra on a Complete Intersection, with an Application to Group Representations." Transactions of the American Mathematical Society 260, no. 1 (1980): 35-64.
Which apparently proves the following theorem: any polynomial with no constant or linear terms can be factored as a product of matrices with no constant terms.
At 11:50 into the video, they comment there is even an algorithm to produce such a factorization.
Unfortunately I do not know enough math to understand the paper and how it relates to what they discussed in the video. I am interested in learning more. Is there a simpler discussion or reference for the algorithm anywhere? Or if it is simple enough, would someone be kind enough to explain it here?
If that is too complex, would it be possible to comment on exactly what properties are proved? For instance, is the factorization unique? Or is it possible to choose the form to some extent?