Matrix Analysis and Computations

Matrix Analysis and Computations is a graduate-level textbook that combines several topics from matrix analysis (matrix factorizations, numerical range, nonnegative and M-matrices in chapters one through four) with coverage of classical and Krylov space methods for the iterative solution of linear systems of equations in chapters five through seven.  The final chapter in the book covers the specialized topic of saddle point systems.  This is an unusual combination of topics that do not seem to be well connected.

 

The book is written in definition-theorem-proof style with very little discussion between the proofs.  There are very few examples or applications in the book.  Rather, the focus is on proving theorems in matrix analysis and the convergence of iterative methods for the solution of linear systems of equations.  The proofs of some propositions are left as exercises.  There is no discussion of computational complexity, memory hierarchies, or parallel algorithms.

 

Compare this book with Matrix Analysis by Roger Horn and Charles Johnson, Yousef Saad's Iterative Methods for Sparse Linear Systems, or David Watkins' Fundamentals of Matrix Computations.  Each of these books is more accessible and thorough in its coverage of the respective topics.  In comparison, the authors of Matrix Analysis and Computations have tried to squeeze too much into a single volume.  

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Brian Borchers is a professor of mathematics at New Mexico Tech.