4 ms·
That has less to do with some mysterious fact about eigenvectors, and more to do with the fact that linear algebra is incredibly incredibly well studied and und
by tw1010 9y ago
That has less to do with some mysterious fact about eigenvectors, and more to do with the fact that linear algebra is incredibly incredibly well studied and understood; with octopus arms sticking into every special case of applied mathematics imaginable. Give it a few decades and "everything" will be some other (today obscure) mathematical idea.
- tzahola 9y agoA category maybe? Wait a sec... everything already is a category.
- cscurmudgeon 9y agoAnd a set.
- dboreham 9y agoI thought it was known that not everything is a set?
- johncolanduoni 9y agoThere’s more than one way to define what a set is, and many of these definitions always allow you to postulate there exist further collections that cannot meet that definition of a set ad infinitum. Most importantly ZFC (which is at least in theory the formal basis for the vast majority of mathematics) has this property, and so do many extensions like NBG and TG.
- chestervonwinch 9y agoof course - it's the empty set :)
- deleted 9y ago[deleted]
- comicjk 9y agoAt least one part of the universe really does seem to prefer linear algebra: quantum mechanics. Every quantum state is a linear combination of eigenvectors, and that's unlikely to change within a few decades (it would be a revolution analogous to the discovery of quantum mechanics itself).
- tw1010 9y agoI would argue that the reason QM is expressed in linear algebra is also not a mysterious property of LA, but because that is what was part of the physicists curriculum when the field was developed.
- comicjk 9y agoPhysicists would have preferred to do it with calculus -indeed, intro QM is usually taught with calculus, not with linear algebra. But you can't go very far without talking about operators and eigenstates. Linear algebra fits the underlying physical phenomenon. People have tried to develop nonlinear versions of quantum mechanics, with no success so far. In particular, if quantum mechanics was not exactly linear, then a quantum computer would be able to solve NP-complete problems in polynomial time! To get an intuition of why quantum mechanics looks the way it does, Scott Aaronson is extremely helpful: https://www.scottaaronson.com/democritus/lec9.html https://www.scottaaronson.com/democritus/lec9.html
- qubex 9y agoYou can also express QM/QFT in terms of matrix algebra (S-Matrix formalism).
- comicjk 9y agoThat's a subset of the linear algebra used in quantum mechanics, no? Any transformation between QM states can be expressed as a unitary matrix - this is not a particular characteristic of scattering phenomena with S-matrices. The quantum states are still vectors, as in the rest of QM.