Linear Algebra
Bibliographic details
Summary
initially conceived so that certain topics, as suggested by reviewers, could be incorporated to make the book useful to a wider readership. Apart from these new topics, select portions of the first edition, mainly in Chapters 3, 6 and 7, have been rewritten for greater clarity for the present edition. Moreover, material in this edition is arranged in such a way that due importance can be given to real symmetric matrices in the initial chapters only. As a result, it has been possible to present the important result about the orthogonal diagonalizibility of real symmetric matrices in Chapter 5 (where diagonalizable operators are discussed). This removes a major drawback of the first edition, where one had to wait till Chapter 8 to obtain the same result as a consequence of the theory of self-adjoint operators in general inner product spaces. The following are the new additions in this edition: LU factorization (Section 2.6; permutation matrices needed for this section are introduced in a new subsection in Section 2.3), orthogonality in Rn with respect to the standard dot product(Section 3.7; this section also deals with orthogonal and unitary matrices and Gram Schmidt process), orthogonal diagonalizibility of real symmetric matrices (Section 5.3, in a new subsection), groups preserving bilinear forms such as orthogonal, pseudoorthogonal and symplectic groups (Section 7.5). A new section 1.7 contains definitions and examples of groups, rings and fields for the benefit of readers not familiar with these concepts. Because of these new material, we have to omit the last chapter of the first edition on selected topics in this edition to keep the book to a reasonable length; the omitted chapter, along with a new section on difference equations and recurrence relations, will be uploaded to the website of the book. However, two important topics from the omitted chapter are included in this edition at appropriate places. The first one deals with conic sections as an application of real quadratic forms (Section 5.7) and the other briefly discusses positive definite and positive semi definite matrices (in Section 7.4). The topic of rational canonical forms in Chapter 6 of the first edition is shifted to the website. The other major changes in this edition are as follows: a new proof of the result that the characteristic and the minimal polynomials of a linear operator have the same irreducible factors, which does not use field extensions (Proposition 5.6.6), correction of the proof of existence of SN decomposition (Proposition 6.2.3) and a simpler proof of Sylvester’s law of inertia (Theorem 7.4.8). Some incorrect statements in the first edition are either corrected or removed for this edition. Some new exercises are also added in this edition. While preparing this edition, I have had many fruitful discussions with several of my colleagues, especially with Prof. A. K. Das of North Eastern Hill University and Prof. R. Shukla of Allahabad University; their valuable suggestions helped me in formulating the contents of several new topics. P