150 years old and still alive: eigenproblems

Abstract

Linear eigenproblems continue to be an important and highly relevant area of research in numerical linear algebra. Therefore, it should be no surprise that numerical algorithms for eigenproblems are among the oldest known in the modern literature. In 1846, exactly 150 years ago, Jacobi [33] wrote his famous paper on the computation of solutions for the problem Ax = x, with AT = A. Strictly speaking, this is not correct, since matrix notation was unknown at that time. So Jacobi formulated the problem in terms of systems of equations, written out elementwise. His paper contains computational elements that are still in use, like the plane rotations for making a system more diagonally dominant, and the (Gauss)-Jacobi iteration method for diagonally dominant systems. These elements were combined in his approach for the computation of eigenvalues, together with a clever way of setting up a correction equation for eigenvalue and eigenvector approximations.