I learned a lot about linear algebra today! I learned that if you exponentiate a matrix (Yes, matrix exponentiation; if this makes you uncomfortable, think about the Taylor series for exponentiation. Do that with a matrix.), the determinant of the resulting matrix is the exponential of the trace of the exponent matrix. So if you need unit-determinant matrices, you can make them by multiplying together exponents of traceless matrices.
Why do I care about all this? Because Jason Hunt (Flatiron), Adrian Price-Whelan (Flatiron), and I realized yesterday that we need to make some of our transformation matrices volume-preserving. This is in our MySpace project for ESA Gaia EDR3 that finds a data-driven, transformation of phase-space to emphasize velocity structure. And I learned that Jax (the simple auto-differentiation tool for numpy and scipy) knows about matrix exponentiation.
I give thanks to Soledad Villar (JHU) for these insights about linear algebra.
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