Matrix Algebra: Exercises and Solutions

This book contains over 300 exercises and solutions covering a wide variety of topics in matrix algebra. They can be used for independent study or in creating a challenging and stimulating environment that encourages active engagement in the learning process. Thus, the book can be of value to both teachers and students. The requisite background is some previous exposure to matrix algebra of the kind obtained in a first course. The exercises are those from an earlier book by the same author entitled "Matrix Algebra From a Statistician's Perspective". They have been restated (as necessary) to stand alone, and the book includes extensive and detailed summaries of all relevant terminology and notation. The coverage includes topics of special interest and relevance in statistics and related disciplines, as well as standard topics. The overlap with exercises available from other sources is relatively small. David A. Harville is a research staff member in the Mathematical Sciences Department of the IBM T.J. Watson Research Center. Prior to joining the Research Center, he served ten years as a mathematical statistician in the Applied Mathematics Research Laboratory of the Aerospace Research Laboratories at Wright-Patterson Air Force Base, Ohio, followed by twenty years as a full professor in the Department of Statistics at Iowa State University. He has extensive experience in linear statistical models, which is an area of statistics that makes heavy use of matrix algebra, and has taught (on numerous occasions) graduate-level courses on that topic. He has authored over 70 research articles. His work has been recognized by his election as a Fellow of the American Statistical Association and the Institute of Mathematical Statistics.

This collection of exercises and their solutions will be a useful reference for students and researchers in matrix algebra. It will be of interest to mathematicians and statisticians.

1 Matrices.- 2 Submatrices and Partitioned Matrices.- 3 Linear Dependence and Independence.- 4 Linear Spaces: Rowand Column Spaces.- 5 Trace of a (Square) Matrix.- 6 Geometrical Considerations.- 7 Linear Systems : Consistency and Compatibility.- 8 Inverse Matrices.- 9 Generalized Inverses.- 10 Idempotent Matrices.- 11 Linear Systems: Solutions.- 12 Projections and Projection Matrices.- 13 Determinants.- 14 Linear, Bilinear, and Quadratic Forms.- 15 Matrix Differentiation.- 16 Kronecker Products and the Vec and Vech Operators.- 17 Intersections and Sums of Subspaces.- 18 Sums (and Differences) of Matrices.- 19 Minimization of a Second-Degree Polynomial (in n Variables) Subject to Linear Constraints.- 20 The Moore-Penrose Inverse.- 21 Eigenvalues and Eigenvectors.- 22 Linear Transformations.- References.