Finite Matrices and Their Nonsingularity

Abstract

Nonsingularity of matrices plays a vital role in the solution of linear systems, matrix computations, and numerical analysis. A large variety of problems arising in computational fluid mechanics, fluid dynamics, and material engineering that are modelled using difference equations or finite element methods require that the matrix under consideration is nonsingular, in order for the numerical schemes to be convergent. In spite of the large scale availability of excellent software for the computation of eigenvalues, there is always a growing need for new results on invertibility of matrices and inclusion regions of spectra of matrices. This is true especially due to the fact that in practical problems, matrices are dependent on parameters. Further, bounds for eigenvalues of finite matrices usually lead to derivation of bounds for the spectra of infinite matrices. Due to these reasons, discovering new sufficient conditions for matrix invertibility and eigenvalue inclusion regions are very relevant even today.