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Matrix Analysis and Applied Linear Algebra

Carl D. Meyer
Publication Date: 
Number of Pages: 
Hardcover with CDROM
[Reviewed by
Joel Foisy
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When I first saw the title of this book, I expected a linear algebra text that would rely heavily on supplemental computer lab work involving MAPLE or Matlab. I was somewhat surprised to find that it did not. In the author's preface, he states "Computing projects help solidify concepts, and I include many exercises that can be incorporated in to a laboratory setting. But my goal is to write a mathematics text that can last, so I don't muddy the development by marrying the material to a particular computer package or language." Indeed the organization and content of the book, on the surface, appear quite theoretical. The chapter titles are Linear Equations, Rectangular Systems and Echelon Forms, Matrix Algebra, Vector Spaces, Norms, Inner Products and Orthogonality, Determinants, Eigenvalues and Eigenvectors, and Perron-Frobenius Theory. One could easily teach a traditional one-semester linear algebra course out of this text.

The author does weave in practical considerations into each chapter. For example, in the first chapter, he goes into detail about algorithmic differences in Gauss versus Gauss-Jordan, and he goes into even greater detail about how to minimize rounding errors when performing Gaussian elimination on a computer. There is a section devoted to analyzing systems for which some small perturbation can produce relatively large changes in the exact solution. He returns to this topic in chapter 3 in a section entitled "Inverses of Sums and Sensitivity," introducing the conditioning number, which measure the degree of sensitivity of a system to small perturbations. He later returns to this topic in the inner product chapter, so that it may be explored in more depth. This type of spiral approach is common in this text, as is the use of many cross-references. I like how well thought out and organized this book is.

In terms of content, I found the book to be somewhat similar to Gilbert Strang's Linear Algebra and Its Applications.. Meyer's style is a bit more formal than Strang's. Meyer takes care to provide many historical footnotes, giving the topics a bit more context than the usual textbook. There are a good variety of exercises, ranging from routine and computational to more difficult, and many exercises foreshadow topics that will be treated later. This text uses some slightly more sophisticated approaches than most elementary linear algebra books on the market. For example, an elementary matrix is first defined as a matrix of the form I - uvT, where U and v are n x 1 columns such that vT u does not equal 1.

Of course, those who teach applied linear algebra courses will want to examine this book. I would recommend that anyone who teaches a course in linear algebra consider this text. Those who choose not to adopt this text would still find it a handy reference — a good reminder of some of the practical issues of linear algebra that working scientists must consider. In my opinion, the stronger the students in the course, and the longer they are exposed to the text (it would be best in a two semester sequence) the better they will appreciate this book and its spiral development of ideas.

See also the author's home page for this book.

Joel Foisy ( is assistant professor of mathematics at SUNY Potsdam in Potsdam, NY.

Chapter 1: Linear Equations. Introduction; Gaussian Elimination and Matrices; Gauss-Jordan Method; Two-Point Boundary-Value Problems; Making Gaussian Elimination Work; Ill-Conditioned Systems; Chapter 2: Rectangular Systems and Echelon Forms. Row Echelon Form and Rank; The Reduced Row Echelon Form; Consistency of Linear Systems; Homogeneous Systems; Nonhomogeneous Systems; Electrical Circuits; Chapter 3: Matrix Algebra. From Ancient China to Arthur Cayley; Addition, Scalar Multiplication, and Transposition; Linearity; Why Do It This Way?; Matrix Multiplication; Properties of Matrix Multiplication; Matrix Inversion; Inverses of Sums and Sensitivity; Elementary Matrices and Equivalence; The LU Factorization; Chapter 4: Vector Spaces. Spaces and Subspaces; Four Fundamental Subspaces; Linear Independence; Basis and Dimension; More About Rank; Classical Least Squares; Linear Transformations; Change of Basis and Similarity; Invariant Subspaces; Chapter 5: Norms, Inner Products, and Orthogonality. Vector Norms; Matrix Norms; Inner Product Spaces; Orthogonal Vectors; Gram-Schmidt Procedure; Unitary and Orthogonal Matrices; Orthogonal Reduction; The Discrete Fourier Transform; Complementary Subspaces; Range-Nullspace Decomposition; Orthogonal Decomposition; Singular Value Decomposition; Orthogonal Projection; Why Least Squares?; Angles Between Subspaces; Chapter 6: Determinants. Determinants; Additional Properties of Determinants; Chapter 7: Eigenvalues and Eigenvectors. Elementary Properties of Eigensystems; Diagonalization by Similarity Transformations; Functions of Diagonalizable Matrices; Systems of Differential Equations; Normal Matrices; Positive Definite Matrices; Nilpotent Matrices and Jordan Structure; The Jordan Form; Functions of Non-diagonalizable Matrices; Difference Equations, Limits, and Summability; Minimum Polynomials and Krylov Methods; Chapter 8: Perron-Frobenius Theory of Nonnegative Matrices. Introduction; Positive Matrices; Nonnegative Matrices; Stochastic Matrices and Markov Chains.