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Linear algebra. (English) Zbl 0904.15001

Pure and Applied Mathematics. A Wiley-Interscience Series of Texts, Monographs & Tracts. New York, NY: John Wiley & Sons. xiv, 250 p. (1997).
This is an excellent addition to the literature. It is based on a course for entering graduate students at the Courant Institute of New York University, and clearly the author thinks this is “linear algebra every applied mathematician and physicist should know”. Little previous knowlege of the subject is assumed but each chapter goes remarkable far. The style is succinct, informal and pleasant. The level is high but a serious student will be able to fill in the details.
The classic topics of real and complex spaces and matrices, including duality and the spectral theory of selfadjoint maps, are covered in the first eight chapters. Pages 93-220 hold the following chapters, each short but rich: 9. Calculus of vector and matrix valued functions, 10. Matrix inequalities, 11. Kinematics and dynamics, 12. Convexity, 13. Duality theorem, 14. Normed linear spaces, 15. Linear mappings between normed spaces, 16. Positive matrices, and 17. How to solve systems of linear equations. There are eight very useful appendices, including Paff’s theorem, symplectic matrices, tensor products and fast matrix multiplication.
Here are a few of the less well-known topics. In Chapter 9 is Rellich’s theorem, that the eigenvalues of \(A(t)\) are an analytic function of \(t\) when \(t\to A(t)\) is an analytic function into real positive definite matrices. Von Neumann and Wigner’s explanation of “avoidance of crossing” is given. This phenomenon, discovered in the early days of quantum mechanics, is that when real symmetric matrices \(B\) and \(M\) are chosen randomly and one calculates the eigenvalues of \(A(t)= B+ tM\) on a sufficiently dense set, a pair of adjacent eigenvalues will appear to be converging but will diverge just before they collide. In chapter 10, three proofs are given for the concavity of \(\log(\text{det}(A))\) for real positive definite \(A\). Various eigenvalue inequalities are there also, including the Wielandt-Hoffman theorem. Chapter 12 includes the theorems of Birkhoff and Helly. Chapter 17 is about iterative methods such as using Chebychev polynomials, and an optimal three term recursion is developed.
There are minor blemishes. It would be good to emphasize finite-dimensionality at times, for example in Chapter 8, where the author remarks that the unit sphere is compact, and in Chapter 2 on duality, where he states \(X\) is finite-dimensional in the first theorem but not in subsequent results when that hypothesis is still needed. The exercises are good but too few by far. There are numerous typographical errors. In particular, the names of Roger Horn and Charles R. Johnson are misspelled in the bibliography, and exercise 1 on page 143 say “…form (11)…condition (5)…”.

MSC:

15-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to linear algebra
53A45 Differential geometric aspects in vector and tensor analysis
15A45 Miscellaneous inequalities involving matrices
15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
26B12 Calculus of vector functions
53A17 Differential geometric aspects in kinematics
15B48 Positive matrices and their generalizations; cones of matrices
15A06 Linear equations (linear algebraic aspects)
65F10 Iterative numerical methods for linear systems
15A18 Eigenvalues, singular values, and eigenvectors
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