Page 8 - Matrices theory and applications

Preface

The study of matrices occupies a singular place within mathematics. It
is still an area of active research, and it is used by every mathematician
and by many scientists working in various specialities. Several examples
illustrate its versatility:

• Scientific computing libraries began growing around matrix calculus.
As a matter of fact, the discretization of partial differential operators
is an endless source of linear finite-dimensional problems.
• At a discrete level, the maximum principle is related to nonnegative
matrices.
• Control theory and stabilization of systems with finitely many degrees
of freedom involve spectral analysis of matrices.
• The discrete Fourier transform, including the fast Fourier transform,
makes use of Toeplitz matrices.
• Statistics is widely based on correlation matrices.

• The generalized inverse is involved in least-squares approximation.
• Symmetric matrices are inertia, deformation, or viscous tensors in
continuum mechanics.
• Markov processes involve stochastic or bistochastic matrices.

• Graphs can be described in a useful way by square matrices.