I don't know any abstract algebraists personally, which is why I'm asking this question here.
Let $(R,*)$ be a commutative $*$-ring where $*:R \to R$ is an involution. Each conjecture below is stated individually for each commutative $*$-ring.
Consider the following three notions:
Naive SVD: A matrix $M$ has a naive SVD if it can be factorised as $M = U S V^*$ where $S$ is a Hermitian and diagonal matrix.
Over some rings, many matrices, including some invertible ones, may not have naive SVDs. For this reason, we introduce two different notions of singular block decomposition or SBD.
Strong $B$-SBD: A matrix $M$ has a strong $B$-SBD if $M$ can be factorised as $M = U S V^*$ where $S$ is the direct sum of matrices in $B$. The multisubset of matrices in $B$ which forms $S$ must be unique, meaning that if $USV^* = U' S' (V')^*$ then $S = S'$ (in the sense that every matrix in $B$ which occurs in $S$ also occurs in $S'$ the same number of times). Additionally, we require $B$ to consist of only Hermitian matrices.
I conjecture that there exists a set $B$ of Hermitian matrices such a matrix has a strong $B$-SBD if and only if it has a polar decomposition.
Weak $B$-SBD: This is the same as the strong $B$-SBD except we don't require $B$ to consist of Hermitian matrices.
I conjecture that there exists a set $B$ such that every matrix has a weak $B$-SBD.
My question is: Have these notions been studied? Are these conjectures easy to prove? I imagine the Axiom of Choice can help select the matrices in $B$, but I haven't fleshed this out.
