Preface

The present textbook contains a collection of six high-quality articles. In particular, this book is devoted to Linear Mathematics by presenting problems in Applied Linear Algebra of gen‐ eral or special interest. In what follows, we present a short summary focusing on the key concepts of each chapter.

In Chapter 1, the authors investigate Krylov subspaces spectral (KSS) methods and exponen‐ tial propagation iterative (EPI) methods. Combination of KSS with EPI methods is presented in Section 5. Especially, the combination of KSS with EPI methods is presented for the pur‐ pose of solving systems of ODEs that are obtained through spatial discretization of nonlin‐ ear PDEs, or systems of nonlinear PDEs, defined on rectangular domains with periodic, homogeneous Dirichlet, or homogeneous Neumann boundary conditions. Analysis of sever‐ al numerical tests is presented in Section 6.

Chapter 2 demonstrates two methods for direct solutions of inverse eigenvalue problems. Both methods can be applied to matrices of all types. Also, the methods are capable of pro‐ ducing multiple solutions, but careful care must be taken in setting up the forward model‐ ling problem because it has a large effect on the order and solvability of the resulting inverse problem.

In Chapter 3, the author focuses on the fundamental importance of eigenvalue problems. The linear as well as the quadratic eigenvalue problem is studied thoroughly. Additional emphasis is given on the QR algorithm, the Rayleigh quotient iteration for linear problems of eigenvalues, and the linearization and minmax characterization of quadratic eigenvalues problems. Finally, a new Rayleigh functional is introduced for gyroscopically stabilized sys‐ tem that enables complete minmax characterization of eigenvalues.

Chapter 4 investigates the nonnegative inverse elementary divisors problem. The elementa‐ ry divisors of a given matrix A are the characteristic polynomials of the Jordan blocks of the Jordan canonical form of A. In the nonnegative inverse elementary divisors problem, we try to find necessary and sufficient conditions for the existence of a nonnegative matrix with prescribed elementary divisors. Problems of this kind are closely related to inverse eigenval‐ ue problems where we try to construct a matrix from some spectral information.

In Chapter 5, the authors study scattering problems for the nonlinear Schrödinger equation

$$-\Delta \boldsymbol{\omega} = \boldsymbol{h}(\boldsymbol{\chi}, |\boldsymbol{\omega}| \, | \boldsymbol{\omega} = \boldsymbol{k}^2 \boldsymbol{\omega}, \ \boldsymbol{\chi} \in \mathbb{R}^2$$

This chapter follows a very interesting approach with extensive use of Linear Algebra meth‐ ods.

#### XII Preface

Chapter 6 is a combination of Statistical methods with Linear Algebra methods. In particu‐ lar, the author study likelihood ratio test procedures in multivariate linear models focusing on projection matrices.

> **Dr Vasilios N. Katsikis** Department of Economics, National and Kapodistrian University of Athens, Athens, Greece
