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I find it fascinating that we're using an overall linear model (Quantum Physics) of our non-linear reality, particularly the bits that are less linear (Bose Einstein condensates) to approximate linear solutions to non-linear models (flow dynamics, etc) of said non-linear reality.

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To be fair, we're using a linear model (Schrödinger's Equation) but in an infinite dimensional vector space and linear operators on these infinite dimensional vector spaces.

These aren't your grandfather matrices and there are things in infinite dimension that are rather weird, specifically when it comes to eigenvalues, which QM makes extensive use of.

The spectrum of an operator, for example can be a weird mix of discrete values, discrete sets of intervals and/or the entire real line.

The typical paradox that leads to consideration of trace class matrices is usually to consider the trace of the commutator of putatively traceless operators.

The inner space of our reality is linear.

That's an extraordinary (and unsupported, here) claim.

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