This slim text’s goal is to help the reader (primarily first-semester graduate students in various mathematical and scientific disciplines) to understand the sensitivity of matrix computations to errors in the input data.
After a 22-page survey of basic matrix information, the author provides a careful introduction to errors, norms, and questions of sensitivity. Subsequent chapters address her real focus: solutions of linear systems, least squares problems, and singular value decomposition. The author does a nice, no-nonsense job here. For example, Ipsen provides a simple derivation of the QR factorization (A = QR, with Q unitary and R upper triangular) from the Cholesky factorization (M = LL*, where L is lower triangular) of A*A, without invoking algorithms such as the Gram-Schmidt method.
In carrying out her agenda, Ipsen deviates from other treatments in several ways, including a simplified concept of numerical stability in exact arithmetic, a high-level view of algorithms, and the use of complex vectors and matrices throughout. Topics are treated using numerical insight as well as mathematical rigor. Most sections end with exercises ― easier exercises labeled with Roman numerals and more challenging problems denoted by Arabic numerals. These problems provide useful facts.
There are a few minor glitches. For example, on p. 30, the value of the infinity norm of a vector is given several lines before the definition of the norm. On p. 112, several plus signs are missing between subspaces.
Even though numerical analysis is not one of my favorite parts of linear algebra, I found this a neat little book. In addition to its stated purpose, it could also serve to supplement the numerical aspects of an undergraduate linear algebra course. The book is described as suitable for self-study, but the book’s usefulness for this purpose suffers from the lack of solutions to exercises.
Ipsen’s book may not conform to every instructor’s view of what a matrix analysis text should be (see Laub's Matrix Analysis for Scientists and Engineers, reviewed here, for an alternative which does not stress numerical analysis), but it deserves a careful look despite its pink cover.
Henry Ricardo (firstname.lastname@example.org) has retired from Medgar Evers College (CUNY), but continues to serve as Governor of the Metropolitan NY Section of the MAA. He is the author of A Modern Introduction to Differential Equations (Second Edition). His linear algebra text was published in October 2009 by CRC Press.