Matrix analysis and applied linear algebra have long been fundamental tools in mathematical disciplines as well as fertile fields for research in their own right. In this course, we will present classical and recent results of matrix analysis that have proved to be important and useful to various areas. We assume background equivalent to a one-semester elementary linear algebra course and knowledge of rudimentary calculus concepts.
Facts about matrices, beyond those found in an elementary linear algebra course, are necessary to understand virtually any area of mathematical science, it maybe linear and nonlinear partial differential equations, multivariate probability and statistics, optimization, linear programming. Matrix analysis are widely used in engineering fields, such as systems and control, signal and image processing, communications and networks, data analysis, machine learning, artificial intelligence, computer vision, and many more—and are considered to be indispensable tools for modern scientists and engineers. Matrix theory also play key roles in theoretical and applied economics and operations research, to name only a few.
One view of “matrix analysis” is that it consists of those topics in linear algebra that have arisen out of the needs of mathematical analysis, such as multivariable calculus, complex variables, differential equations, harmonic analysis, optimization, and approximation theory. Another view is that matrix analysis is an approach to real and complex linear algebraic problems that does not hesitate to use notions from analysis – such as limits, continuity, convergence or divergence, norms and power series – when these seem more efficient or natural than a purely algebraic approach.
