Handbook of linear algebra (Q2836473)

This book consists of 1904 pages, with many chapters written by many different authors. The second edition contains 500 pages more than the first one. The reviewer of the first edition asked the following questions: a) How complete is this book? b) How useful is it for experts and for those who are not experts?NEWLINENEWLINEThe reviewer of the second edition thinks that the answer to the first question is: `not complete', and to the second question is: `yes'. To give some examples of the reasons for the negative answer to question a), let us mention that the condition number of the Hilbert matrix is not estimated and, more generally, methods for estimating the condition number of large non-symmetric matrices are not given. There is a reference to a paper of \textit{M.-D. Choi} [Am. Math. Mon. 90, 301--312 (1983; Zbl 0546.47007)], in which one can find some formula for the condition number of the Hilbert matrix of an arbitrary order. In the book under review on p. 50-11 one can find just a table of calculated condition numbers of Hilbert matrices of the order not larger than \(13\).NEWLINENEWLINEThe practically important question of stable solving of ill-conditioned (ill-posed) linear algebraic systems is not found in the index. The presentation in Sections 50-4 and 50-5 of the numerically stable solutions of linear algebraic systems is very limited in scope and the example of an ill-conditioned linear algebraic system on p. 50-10 is rather trivial. There is no mentioning of stable methods for solving ill-conditioned linear algebraic systems, for example, the dynamical systems method (see [\textit{A. G. Ramm}, Inverse problems. Mathematical and analytical techniques with applications to engineering. New York, NY: Springer (2005; Zbl 1083.35002); Dynamical systems method for solving operator equations. Amsterdam: Elsevier (2007; Zbl 1245.37012); \textit{A. G. Ramm} and \textit{N. S. Hoang}, Dynamical systems method and applications. Theoretical developments and numerical examples. Hoboken, NJ: John Wiley \& Sons (2012; Zbl 1241.65053); Acta Appl. Math. 111, No. 2, 189--204 (2010; Zbl 1236.65039); Ann. Pol. Math. 95, No. 1, 77--93 (2009; Zbl 1161.47059)])) and other methods.NEWLINENEWLINEHere are a few remarks to help the editor and the authors to improve this handbook that, probably, will be used by many readers since linear algebra is widely used.NEWLINENEWLINEIn the preliminaries, on p. 0-7, the reader sees an example \(\ln (x)=o(x)\) and may think that this is an error if \(x\to 0\). In this example one can easily guess that the author had \(x\to +\infty\) in mind. However, it is always advisable to write the limit to which \(x\) tends.NEWLINENEWLINEOn the same page, a definition of multiset is given, but no definition of a set can be found on this or the other pages of the preliminaries. Therefore, the author's claim that \({1,1,3,-2,-2,-2}\) is an example of a multiset which is not a set is not clear.NEWLINENEWLINEOn p. 2-5 the symbol used for the direct sum is the same as is usually used for the orthogonal sum.NEWLINENEWLINEOn p. 28-2, line 9, in the equality \(p_A(x)=f\) the \(f\), \(p_A(x)\) and \(x\) are not defined.NEWLINENEWLINEIn Section 65, Application to mathematical physics, there is no discussion of applications of linear algebra to wave scattering by many small particles (see [\textit{A. G. Ramm}, Scattering of acoustic and electromagnetic waves by small bodies of arbitrary shapes. Applications to Creating New Engineered Materials. New York: Momentum Press (2013); ``Many-body wave scattering problems in the case of small scatterers'', J. Appl. Math. and Comput., (JAMC), 41, N1, 473--500 (2013); Rep. Math. Phys. 71, No. 3, 279--290 (2013; Zbl 1302.35137); \textit{M. Andriychuk} and \textit{A. G. Ramm}, ``Numerical solution of many-body wave scattering problem for small particles and creating materials with a desired refraction coefficient'', in: Numerical simulations of physical and engineering processes. Wien: InTech. 1--28 (2011)]).NEWLINENEWLINEIn Sections 78 and 79 it would be useful to discuss the nonlinear differential inequalities that extend considerably the classical approach to stability of the solutions to differential equations, see [\textit{A. G. Ramm}, ``Stability of solutions to some evolution problems'', Chaotic Model. Simul. 1, 17--27 (2011); J. Math. Anal. Appl. 396, No. 2, 523--527 (2012; Zbl 1258.34151); Math. Methods Appl. Sci. 36, No. 4, 422--426 (2013; Zbl 1267.34113)].NEWLINENEWLINEIt would be useful to have the list of the authors of the references that are put at the end of each chapter.NEWLINENEWLINEThe above remarks hopefully will help to improve the next edition of this book which is useful to many users of linear algebra.