Edit: I was advised to replace the question with a more specific one.
Coming from a very theoretical background, I'm pretty ignorant about what practical matrix solvers exist. (I have been, and will continue to scour the web for information, but I figured I would get direct and concise answers here as well.)
Currently, what are the most important matrix solvers? (implemented in any software)
I have a rudimentary programming background, and I'm familiar with classical theoretical methods for solving matrix systems. However, I'm well aware computer scientists have been at work finding ingenious ways to solve larger systems more quickly, especially for specialized classes of matrices.
I'm looking at applications of matrix solvers in industry, in particular for use in simulation programs.