A C++ template library for linear algebra: matrices, vectors, numerical solvers, and related algorithms.
Eigen is a self-contained library covering a very broad range of use cases. For example, it covers both dense and sparse objects, and in the dense case, it covers both fixed-size and dynamic-size objects. Moreover it provides linear algebra algorithms, a geometry framework, etc. It has a very nice API for C++ programmers, and it embraces very high performance.