*(English)*Zbl 1122.15001

This Handbook covers linear algebraic things, such as matrices, matrix theory, matrix numerical methods, matrix applications, and the relevant aspects of several fields that have proven beneficial to matrix analysis such as graph theory, combinatorics, optimization, algebra, geometry, etc.

The book is composed of 77 chapters with 99 contributing authors using 4 editors. The chapters are arranged in 5 parts: part 1 on linear algebra, part 2 on combinatorial matrix theory and graphs, part 3 on numerical methods for dense and sparse matrices, part 4 on applications, and part 5 on computational and symbolic software for matrix problems. A glossary of 40 pages, 9 pages of notations, and an index of 56 double columned pages conclude the book.

The book weights in at 2350 gram or 5 lbs 2 ounces. The page numbers start over at 1 in each new chapter. Overall there are well over a thousand pages; one source counts 1216 pages, another 1400. The index refers to chapter and local page numbers as 43–9 for example for page 9 in chapter 43 which is about halfway through the book. So it takes an effort to locate a subject from the index. Once located, each chapter starts with basic definitions and a list of known subject facts with quotes to the literature, followed by examples, extensions, and the chapter bibliography which lists only the most general references and not any original ones. There is no central bibliography. In addition, the editor’s website contains 2 short addenda with 10 corrections at this time.

In an admirable and formidable effort, the Handbook covers large swaths of linear algebra, also known as matrix theory, as well as matrix computations and applications in an encyclopedic way. The list of topics and subjects is enormous and the treatment quite up to date. The book is a welcome addition and will provide useful synopses of many old and new results of its field in a compressed form for years to come. The quality of the chapters is generally very high and the handbook should prove useful to anyone working with math computations and matrices. It represents a historic achievement in “handbooking” for mathematics.

How complete is this book? How useful is it for experts and for those who are not? The answers vary. The chapter bibliographies and listed “facts” are often not exhaustive; some of the chapter extensions are more encompassing in their scope and outlook than others which read more like extended bibliographies of their authors, and yet others do not mention essential results of their subject at all. This – unfortunately – will irritate experts and keep novices from grasping the affected subjects comprehensively. Thus the reader should beware and search the original literature as well. The applications part 4 is especially thin. There are no applications to engineering, none to number theory. Regarding missing subject matter, indefinite inner product spaces are not treated, nor are matrix commutators; matrix pencils receive limited treatment, while canonical pair forms are completely missing; there is no mention of bifurcation phenomena or stiffness of dynamical systems, and so forth. Editing is sloppy in parts. For example, inside the glossary of 40 pages there is one full page dedicated to terms starting with “partial ...”; these all occur on only 9 pages inside chapter 35.

These reservations must in no way detract from the huge achievement and effort put forth so splendidly by over 100 researchers and editors here. It would be foolish from now on for anyone trying to uncover anything linear algebraic or matricial not to search the Handbook!

Yet I place my hope into a second, a revised edition with continuous page numbering, chapters of more uniformly wide breadth and depth, and a unified bibliography which also contains original papers.

##### MSC:

15-00 | Reference works (linear algebra) |

65Fxx | Numerical linear algebra |

05C50 | Graphs and linear algebra |

68W30 | Symbolic computation and algebraic computation |