TITLE: Modern developments in finite-dimensional linear algebra
QUESTION [21 upvotes]: 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.

REPLY [6 votes]: Since you lowered the level to Weyl's Inequalities (1912), it is worth mentionning the improvements of these inequalities made by Ky Fan, Lidskii and others. They culminated in a much involved conjecture by A. Horn (1961), eventually proved by Knutson & Tao on the turn of the century.