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Many matrix decompositions - like the Jordan Normal Form, the SVD, the spectral theorem, the Takagi decomposition - have the property that they express a matrix $M$ as the form:

$$M = A D B$$ where $D$ is a block-diagonal matrix, whose block entries are unique up to permutation. I believe that there is a unified scheme for explaining this, but I'm wondering what results in quiver representation theory can be used to show uniqueness results like this. I'm also OK with the representation theory of *-algebras.

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    $\begingroup$ What are the reasons to believe that quiver and/or *-algebras representations have anything to do with this? $\endgroup$
    – Andrei Smolensky
    May 1 at 7:32

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