Modern developments in finite-dimensional linear algebra 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.
 A: 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.
A: This is a borderline suggestion, both in terms of how "major" it is and timing (does 1931 count as "early" 20th century?), but there is the Gershgorin circle theorem.
A: Also a borderline suggestion since it is rather multilinear than just linear: Recent progress on low rank tensor approximation for all kinds of different applications within mathematics. A list of applications from this preprint includes


*

*approximation of multidimensional integrals

*electronic structure calculations

*solving stochastic or parameter dependent PDEs

*approximating Green's functions in high dimensions

*solving Boltzmann-type equations or high-dimensional Schrödinger equations

*rational approximation problems

*computational finance

*multivariate regression and machine learning. 

A: From Wikipedia: "The linear-programming problem was first shown to be solvable in polynomial time by Leonid Khachiyan in 1979, but a larger theoretical and practical breakthrough in the field came in 1984 when Narendra Karmarkar introduced a new interior-point method for solving linear-programming problems."
A: I would say the theory of quivers and in particular Gabriel's theorem on finite representation type and its extensions to tame type. Representations of quivers are essentially linear algebra problems in a different language. For instance Jordan canonical form is the description of indecomposable reps of a quiver with one vertex and a loop. In general things like the classification of two endomorphisms of vector spaces, matrix pencils and the n-subspace problem are all problems in the rep theory of quivers. The intro to the book of Gabriel-Roiter says more. 
Added. A quiver is a directed multigraph, often assumed finite in this context. A representation of a quiver Q is an assignment of a vector space to each vertex and a linear transformation to each edge from the vector space at its source to the vector space if its target. Isomorphisms are isomorphisms of vertex spaces making commuting squares with the edge linear transformations. There is a fairly straightforward notion of direct sum and hence indecomposable rep. Finite rep type means finitely many isoclasses of indecomposables, tame type essentially means indecomposables come in 1-parameter families (plus finitely many exceptions) if you fix the dimensions of the vertex spaces. Wild means its representation theory contains that of all finite dinensional (and hence all finitely generated) algebras. In particular the first order theory is undecidable. Only finite, tame and wild occur. 
A: Definitely, some items on the top of my list are:


*

*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.

*The resolution of Horn's conjecture (see this nice summary article by R. Bhatia, which also mentions several other nice connections)

*Randomised linear algebra and progress on fast solutions to linear systems (see e.g., the very readable summary in N. Vishnoi's web book)

*Advances in quantum information theory? Though I don't know how much of that I would push into just linear algebra

*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).

A: Just putting the references asked for by Timur:


*

*J. M. Landsberg, "The border rank of the multiplication of $ 2\times 2$ matrices is seven", J. American Math. Soc. 19 (2006), 447-459.

*J. M. Landsberg and G. Ottaviani "New lower bounds for the border rank of matrix multiplication", 2011 preprint.

