Are there any major fundamental results in finite-dimensional linear algebra discovered after early XX century? Fundamental in the sense of non-numerical (numerical results, of course, are still interesting and important); and major in the sense of something on the scale of SVD or Jordan normal form. 


(EDIT) As several commenters observed, using Jordan normal form as a benchmark sets the bar way too high. Let's try lowering it to Weyl's inequality.



Modern developments in finite-dimensional linear algebra
Definitely, some items on the top of my list are:


1. Random matrix theory --- both asymptotic and non asymptotic; including things like semi-circular law, circular law, and so on. Check out Terry Tao's blog for very nice summaries.
2. The resolution of Horn's conjecture (see this [nice summary article by R. Bhatia](http://math.ucr.edu/~jdolan/schubert2.pdf), which also mentions several other nice connections)
3. Randomised linear algebra and progress on fast solutions to linear systems (see e.g., the very readable summary in [N. Vishnoi's web book](http://research.microsoft.com/en-us/um/people/nvishno/Site/Lxb-Web.pdf))
4. Advances in quantum information theory? Though I don't know how much of that I would push into just linear algebra
5. Not advances in linear algebra itself, but the gigantic success of basic linear algebra in new areas (machine learning, information retrieval, etc., e.g., Google's PageRank method).


