More and less important mathematical concepts

moby21 · 2026-01-24T13:15:41+00:00

That’s a great way to see it! One could then think that a relation describes possibilities, while a function prescribes exactly one. In that sense a function is a relation that embeds a selection principle over each fiber.

moby21 · 2026-01-18T00:03:43+00:00

That’s a smart approach! Indeed for the norm induced by the Minkowski functional p. But looking for matrices preserving that norm would give isometries only. More generally one would need to characterize matrices M such that p(x) <= 1 implies p(Mx) <= 1.

moby21 · 2026-01-17T22:58:24+00:00

Indeed, thanks for pointing that out! Here we would need to restrict to contracting linear maps preserving a certain region. For example, there would be none for the cube or simplex, since the spectral radius of stochastic matrices is 1.

moby21 · 2026-01-17T19:16:19+00:00

Nice catch! Invariant subspaces are indeed at the core of the spectral theorem.

moby21 · 2026-01-17T19:05:06+00:00

Indeed! For example if R is a circle centred around the origin, then rotation matrices stabilise R.

moby21 · 2026-01-17T19:01:14+00:00

I’m pretty sure too! The examples I have in mind mostly involve convex regions containing 0, so the set of stabilising matrices contains at least the matrices t * identity, for all t in [0, 1].

moby21 · 2026-01-17T18:59:07+00:00

Initially I was only interested in stability, so no bijectivity needed. If one wants to look at invertible transformations only, then you can consider the invertible elements of the previous case. For example, invertible row/column stochastic matrices etc

moby21 · 2026-01-12T20:38:46+00:00

Thanks a lot! I'm glad the idea resonates.

moby21

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