The Schur complement and its applications. (English) Zbl 1075.15002

Numerical Methods and Algorithms 4. New York, NY: Springer (ISBN 0-387-24271-6/hbk). xvi, 295 p. (2005).

Given a square matrix \(A=(A_{ij}), i,j=1,2\) with diagonal blocks \(A_{11} \in \mathbb C^{r,r}, A_{22} \in \mathbb C^{n-r,n-r}\), the matrix \(A/A_{11}= A_{22}-A_{21}A_{11}^{-1}A_{12} \) is called the Schur complement of \(A\) w.r.t. \(A_{11}\). As \(A/A_{11}\) appears after eliminating the first \(r\) components of \(x\) in the system \(Ax=y\), it is obvious that this mapping \(A \rightarrow A/A_{11}\) is important and has many applications in matrix analysis, statistics, numerical analysis and other areas.
The book consists of eight chapters, each written by experts in their field, devoted to certain aspects and applications of the Schur complement. They can be read independently of each other.
Chapter 0: Historical introduction: Issai Schur and the early development of the Schur complement.
Chapter 1: Basic properties of the Schur complement.
Chapter 2: Eigenvalues and singular values of Schur complements.
Chapter 3: Block matrix

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techniques.
Chapter 4: Closure properties.
Chapter 5: Schur complements and matrix inequalities: Operator-theoretic approach.
Chapter 6: Schur complements in statistics and probability.
Chapter 7: Schur complements and applications in numerical analysis.
The book can serve as a research reference, as it contains many new results and results not yet appeared in books. The articles contain thorough expositions, so they can be understood by anyone having a good knowledge of linear algebra

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.

Reviewer: Ludwig Elsner (Bielefeld)

MSC:

15-02	Research exposition (monographs, survey articles) pertaining to linear algebra
62-02	Research exposition (monographs, survey articles) pertaining to statistics
01-02	Research exposition (monographs, survey articles) pertaining to history and biography
15-03	History of linear algebra
65-02	Research exposition (monographs, survey articles) pertaining to numerical analysis
15A18	Eigenvalues, singular values, and eigenvectors
15A45	Miscellaneous inequalities involving matrices
65F30	Other matrix algorithms (MSC2010)
01A60	History of mathematics in the 20th century

Keywords:

Schur complements; statistics; numerical analysis; matrix analysis; textbook; Issai Schur; eigenvalues; singular values; block matrix; matrix inequalities