Chapter 4 starts out with a physics motivation, as well as a mathematical statement of the problems that will be tackled in later sections. An extensive section discusses the conditioning of linear-algebra problems: borrowing ideas and examples from matrix perturbation theory, this teaches students what to look out for. Roughly half of the remaining chapter is dedicated to the solution of linear systems of equations, employing methods of varying sophistication: Gaussian elimination, LU decomposiion, pivoting, and the Jacobi iterative method. The second half addresses the eigenvalue problem, again with a variety of methods, including the power method, eigenvalue shifting, and the QR method. Crucially, this includes explicit mathematical derivations of these approaches. A brief introduction to the singular value decomposition is also given, including both an existence proof and a programming implementation. The chapter is rounded out by an extensive physics project, which studies the eigenvalue problem of interacting spins, and a problem set. The physics project patiently builds up matrix quantum mechanics, allowing students to tackle problems of increasing difficulty.

The algebraic eigenproblem is the mathematical answer to the physical questions:What are the principal stresses in a solid or fluid and on what planes do they act?What are the natural frequencies of a system?Is the system stable to small disturbances?What is the best basis with respect to which to solve a system of linear algebraic equations with a real symmetric coefficient matrix?What is the best basis with respect to which to solve a system of linear ordinary differential equations?What is the best basis with respect to which to represent an experimental or numerical data set?