Eigen is a C++ template library for linear algebra: matrices, vectors, numerical solvers, and related algorithms.
It supports tiny & large dense, and sparse matrices, fully integrated with one another.
It has no dependency and no binary to compile or link-to.
It takes full advantage of expression templates to provide a great API, and enable many optimizations including temporary removal, and intelligent explicit vectorization (SSE, AltiVec, and NEON).
Performance is on par with the best BLAS implementations.
It provides linear solvers comparable to LAPACK as well as sparse direct and iterative methods with preconditioners.
It keeps in mind the real needs of applications, for example the geometry features needed for 3D graphics.