Applied Linear Algebra, by Noble and Daniels, 3rd Edition, Prentice Hall (1988)

Reviewed by Wiley Williams

Despite the name of this book, it is most appropriate for a second course in Linear Algebra that emphasizes Matrix Analysis. The overarching mathematical theme of the text is matrix decomposition. The discussion often includes a discussion of how modern software is used to solve matix problems.

Unusual features of the book include: Early chapters contain (in addition to standard material) descriptions of block multiplication of matrices, left and right inverses, motivational applications of linear algebra, and an extensive discussion of gaussian elimination and the LU-decomposition. After a treatment of vector spaces, independence, basis and dimension, vector and matrix norms (over both the reals and complexes) are discussed, the Banach Lemma is proved, and ill-conditioning is discussed. Eigenvalues are motivated by the vibrating string problem Three perspectives of eigenvalues and eigenvectors are contrasted: diagonalizing a matrix, decomposing a matrix in a form which makes computing powers easy, and finding a basis so that the matrix of the associated linear transformation is diagonal w.r.t the basis. A chapter on normal matrices and orthogonal diagonalization is included. Orthogonal projections are discussed, including Householder matrices, then the Gram-Schmidt process and its implementation as the QR decomposition, as well as the use of this in solving the least squares problem (which was one of the motivational applications in Chapter 2). This culminates in a development of the singular value decomposition and its use in solving the least squares problem. A chapter on generalized eigenvalues and the Jordan Theorem. A chapter on positive definite matrices, quadratic forms, and conic sections. The text includes some problems to be solved using MATLAB, but an instructor who wants to take full advantage of this technology will need to provide a good deal of supplementation. Also, the problem sets tend to not have enough routine problems or interesting theoretical ones for students to fully explore the concepts involved.