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So far, I have mostly done linear algebra over $\mathbb R$ and $\mathbb{C}$. I know there exist very many methods to solve equation systems for those fields, of which a few are Gaussian Elimination, Gauss-Jordan, Krylov subspace methods and various approaches based on matrix factorizations.

Inspired by this attempt; What determines if a particular method can be used on matrices of a particular field? Can they all be used on any field or is there some extra set of requirements?

EDIT I am more curious about the requirements on the field regarding computational methods to calculate rather than what is required to guarantee the existance of a particular form. It is good to know that Jordan Normal Form will exist for any matrix if we are working on an algebraically closed field. But will the algorithms to calculate it in practice have to be adapted to the field or are the requirements on fields "strong enough" in some sense to ensure the algorithms will always work?

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    This depends on the method. Gaussian elimination works over any field, see here. Jordan normal form only works over algebraically closed fields. – Dietrich Burde Jun 27 '16 at 12:35
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    Indeed, Gaussian elimination works over any field. If permitted, I blow my own trumpet for an example. I'm not an expert, but I think that the development of other methods has been motivated by A) a desire to find something computationally more efficient than the Gaussian elimination for special cases needed in some application, and B) to cater for the situations, where exact arithmetic is impossible (limitations of devices in handling real numbers), and/or we want to control the size of the eventual approximation error. – Jyrki Lahtonen Jun 28 '16 at 07:35
  • Yes I have no problem with such things, I too do it all the time :) I hope it to make it easier to follow the train of thoughts both for myself and for others. It is reassuring that gaussian elimination always works, but I have a bit of a computational focus so I'm always looking for more efficient ways to do things. – mathreadler Jun 28 '16 at 07:46

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