I've noticed some correspondences between some matrix decompositions and monoid isomorphisms (always to some free commutative monoid), in addition to the one I asked about in a previous question:
- Given a $*$-ring $R$, let $H(R,*)$ denote the set of Hermitian matrices over $R$ of all possible dimensions. We can make $H(R,*)$ into a monoid using the direct sum operation (denoted $\oplus$). Let $\sim_\text{US}$ be the equivalence relation denoting unitary similarity; in other words, we have that $A \sim_\text{US} B$ if there exists a unitary matrix $U$ such that $A = UBU^*$. Consider the monoid $H(R,*)/{\sim_\text{US}}$, and observe that the monoid is commutative.
- Given a ring $R$, let $J(R)$ denote the set of square matrices of all possible dimensions over $R$. We can make $J(R)$ into a monoid using the direct sum operation (denoted $\oplus$). Let $\sim_\text S$ be the equivalence relation denoting matrix similarity; in other words, we have that $A \sim_\text S B$ if there exists an invertible matrix $P$ such that $A = PBP^{-1}$. Consider the monoid $J(R)/{\sim_\text S}$, and observe that the monoid is commutative.
Define $*:R \oplus R \to R \oplus R$, where $R$ is a ring (but not necessarily a $*$-ring), and $\oplus$ in this case denotes direct sum of rings, by $(x,y)^* = (y,x)$. We have that $H(R \oplus R,*)/{\sim_\text{US}} \cong J(R)/{\sim_\text S}$.
In the following, a free commutative monoid is always a monoid whose elements are all possible finite multisets whose elements belong to some ambient set. The elements of the ambient set are the generators of the monoid.
An isomorphism between $H(R,*)/{\sim_\text{US}}$ and a free commutative monoid is a spectral theorem over the $*$-ring $R$. And an isomorphism between $J(R)/{\sim_\text S}$ and a free commutative monoid is a Jordan decomposition for the ring $R$. Such isomorphisms appear to exist for many isolated rings, and in fact I've yet to see a ring for which these isomorphisms don't exist.
In the previous question, I defined $M(R,*)/{\sim_\text{UE}}$ in order to generalise the Singular Value Decomposition. The above two monoids are motivated here in a similar way.
Has this been studied in the literature to this degree of generality and for general rings?
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Category-theoretical insights: Observe that $H$, $J$, $M$ and their respective quotients are all functors from the category of $*$-rings to the category of monoids. Also, observe that the isomorphism $H(R \oplus R,*)/{\sim_\text{US}} \cong J(R)/{\sim_\text S}$ is natural in $R$. Observe also that the mapping $M \mapsto \begin{pmatrix}0 & M^* \\ M & 0\end{pmatrix}$, which occurs in numerical linear algebra when computing the SVD, is a natural transformation from $M(R,*)/{\sim_\text{UE}}$ to $H(R,*)/{\sim_\text{US}}$.
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A paper I've produced on this topic: Towards a Singular Value Decomposition and spectral theory for all rings. In the paper, I've replaced $H$, $J$ and $M$ with other notation (unless there are typos where the old notation annoyingly persists).
