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The Jordan normal form and the theory of elementary divisors. I, II. 3rd ed. (Japanese) Zbl 0662.15009
Iwanami Shoten Kiso Sūgaku, 2. Senkei Daisū, ii. Tokyo: Iwanami Shoten. V, VI, 320 p. (1987).
In the first volume of this book the author introduces the Jordan normal form. The following important result is shown: A nondiagonalizable transformation A of a vector space over a perfect field can be uniquely decomposed into a sum \(A=S+N\), where S is semisimple and N is nilpotent such that \(SN=NS\), and S,N are polynomials of A. The second volume consists of four chapters, Chapters 2-5. Chapter 2 introduces some fundamental concepts of matrix polynomials and discusses a generalization of the theory of elementary divisors over a commutative ring such as a PID. Chapter 3 contains some algebraic applications, Chapter 4 and Chapter 5 contain some analytic applications.
The reviewer feels that this book is really first rate, and will serve well both as a text for a standard undergraduate course and as a reference work.
Reviewer: Tong Wenting
15A21 Canonical forms, reductions, classification
15-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to linear algebra
15B33 Matrices over special rings (quaternions, finite fields, etc.)