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Yeah I have been realizing this lately as well. I have written up a bit on this https://github.com/adamnemecek/adjoint/ https://github.com/adamnemecek/adjoint/
by adamnemecek 4y ago
Yeah I have been realizing this lately as well. I have written up a bit on this
https://github.com/adamnemecek/adjoint/ https://github.com/adamnemecek/adjoint/
Check the raw source there’s a bunch of links.
Fixed points, diagonalizations and eigenshit are all the same thing.
- ogogmad 4y agoWhat I'm suggesting doesn't appear to be mentioned there.
- adamnemecek 4y agoThe thing that makes it tick is adjoint and norm.
- elcritch 4y agoInteresting, I learned recently the reason that Galois fields are so important in error correction codes (and encryption) is that they enable invertable matrix operations using integers which makes them easy compute in dedicated hardware. Apparently development of the error correction codes used on CDs wouldn't have been feasible with the computing power of the time without being able to build custom silicon for it.
- JadeNB 4y ago> Fixed points, diagonalizations and eigen… are all the same thing. This is true in a moral sense, but not literally. Fixed points are eigenvectors for a particular eigenvalue \lambda = 1 (together with the 0 vector, which is always fixed but not an eigenvector, by fiat). (Over a field,) eigenvectors corresponding to non-0 eigenvalues are not fixed points, although they lie on fixed lines; and they can be made fixed points by re-scaling the original matrix, if that is convenient. I don't know an obvious way to think of eigenvectors with eigenvalue \lambda = 0 as fixed vectors. A matrix that is diagonalizable has sufficiently many eigenvectors, but a matrix may have eigenvectors even if it is not diagonalizable.
- haskellandchill 4y agoEigenvectors for eigenvalue 0 characterize the null space. The eigenvalue 0 can have a multiplicity of eigenvectors, these eigenvectors span the nullspace.