Matrix Algebra: Exercises and Solutions

Description

This book comprises well over three-hundred exercises in matrix algebra and their solutions. The exercises are taken from my earlier book Matrix Algebra From a Statistician's Perspective. They have been restated (as necessary) to make them comprehensible independently of their source. To further insure that the restated exercises have this stand-alone property, I have included in the front matter a section on terminology and another on notation. These sections provide definitions, descriptions, comments, or explanatory material pertaining to certain terms and notational symbols and conventions from Matrix Algebra From a Statistician's Perspective that may be unfamiliar to a nonreader of that book or that may differ in generality or other respects from those to which he/she is accustomed. For example, the section on terminology includes an entry for scalar and one for matrix. These are standard terms, but their use herein (and in Matrix Algebra From a Statistician's Perspective) is restricted to real numbers and to rectangular arrays of real numbers, whereas in various other presentations, a scalar may be a complex number or more generally a member of a field, and a matrix may be a rectangular array of such entities. Front Matter....Pages i-xxxi Matrices....Pages 1-5 Submatrices and Partitioned Matrices....Pages 7-10 Linear Dependence and Independence....Pages 11-12 Linear Spaces: Row and Column Spaces....Pages 13-18 Trace of a (Square) Matrix....Pages 19-20 Geometrical Considerations....Pages 21-26 Linear Systems: Consistency and Compatibility....Pages 27-28 Inverse Matrices....Pages 29-33 Generalized Inverses....Pages 35-47 Idempotent Matrices....Pages 49-54 Linear Systems: Solutions....Pages 55-61 Projections and Projection Matrices....Pages 63-67 Determinants....Pages 69-78 Linear, Bilinear, and Quadratic Forms....Pages 79-112 Matrix Differentiation....Pages 113-137 Kronecker Products and the Vec and Vech Operators....Pages 139-159 Intersections and Sums of Subspaces....Pages 161-178 Sums (and Differences) of Matrices....Pages 179-207 Minimization of a Second-Degree Polynomial (in n Variables) Subject to Linear Constraints....Pages 209-220 The Moore-Penrose Inverse....Pages 221-229 Eigenvalues and Eigenvectors....Pages 231-250 Linear Transformations....Pages 251-264 Back Matter....Pages 265-273