3 ms·
In practice, all matrices are diagonalizable, because the set of non-diagonalizable matrices is very small. Concretely, a random matrix is almost surely diagona
by xyzzyz 3y ago
In practice, all matrices are diagonalizable, because the set of non-diagonalizable matrices is very small. Concretely, a random matrix is almost surely diagonalizable (ie. with probability 1).
- FooBarBizBazz 3y agoThis "with zero probability" statement hides that there's a probability distribution assumed. It's really saying that any matrix can be perturbed to be diagonalizable. Which is true. But matrices in that set have a way of showing up a lot in models that the human mind conceives. For example, if you model a point mass accelerating without drag, you will end up with a matrix that isn't diagonalizable. If all matrices were diagonalizable, every linear ODE would be solved by a bunch of exponentials and sinusoids; there'd be no polynomials. But we see things like "1/2 a t^2" all the time.
- henrydark 3y agoAlso "any matrix can be perturbed" assumes a normed field, though those equations work over other fields as well