Linear Systems: Basic Properties and Classical Algorithms

Linear systems of algebraic equations lead to the most fundamental problems in computational linear algebra. Such "Ax = b" systems arise throughout science, engineering, and data applications, and as building blocks in algorithms for more sophisticated problems. In this bootcamp talk, we will provide an overview of linear systems and their basic properties (sensitivity of x to changes in A and b), then describe classical methods for their solution: matrix-factorization algorithms based on Gaussian elimination, expedited variants that exploit sparsity or other structure, and iterative methods suitable for large-scale systems. The convergence of these iterative methods can depend on subtle ways on properties of the matrix A (eigenvalue distribution, nonnormality), which we will briefly describe.