A matrix eigenvalue problem considers the vector equation
Here A is a given square matrix, λ an unknown scalar, and x an unknown vector. In a matrix eigenvalue problem, the task is to determine λ's and x's that satisfy (1). Since x = 0 is always a solution for any λ and thus not interesting, we only admit solutions with x ≠ 0.
The solutions to (1) are given the following names: The λ's that satisfy (1) are called eigenvalues of A and the corresponding nonzero x's that also satisfy (1) are called eigenvectors of A.
From this rather innocent looking vector equation flows an amazing amount of relevant theory and an incredible richness of applications. Indeed, eigenvalue problems come up all the time in engineering, physics, geometry, numerics, theoretical mathematics, biology, environmental science, urban planning, economics, psychology, and other areas. Thus, in your career you are likely to encounter eigenvalue problems.
We start with a basic and thorough introduction to eigenvalue problems in Sec. 8.1 and explain (1) with several simple matrices. This is followed by a section devoted entirely to applications ranging from mass–spring systems of physics to population control models of environmental science. We show you these diverse ...