Magnus, Jan R. Linear structures. (English) Zbl 0667.15010 Griffin’s Statistical Monographs and Courses, 42. London: Charles Griffin & Company Ltd.; New York etc.: Oxford University Press. xii, 205 p. £23.50 (1988). By a linear structure the author means the set of all \(m\times n\) real matrices that satisfy a given set of linear restrictions. Thus, symmetric, skew-symmetric, lower triangular, diagonal, circulant and Toeplitz matrices are all examples of linear structures. Each linear structure can be characterized by a particular 0-1 matrix. For example, the commutation matrix \(K_{mn}\) is the mn\(\times mn\) 0-1-matrix such that \(K_{mn}vec A=vec A'\) for all \(m\times n\) matrices A, where vec A is the mn\(\times 1\) vector obtained by stacking the columns of A in order one under the other. (In this case the linear structure is the set of all \(m\times n\) matrices with no constraints.) Another example is the duplication matrix which is the \(n^ 2\times n(n+1)\) 0-1 matrix \(D_ n\) such that \(D_ nv(A)=vec A\) for all \(n\times n\) symmetric matrices A, where v(A) is the \(n(n+1)\times 1\) vector obtained by eliminating all the supradiagonal elements of A from vec A. In the present monograph the author gives a detailed account of the properties of the 0-1 matrices which arise in the characterization of a number of linear structures. He also gives applications to matrix equations, the normal distribution and maximum likelihood. Each chapter (except Chapter 2) contains exercises and bibliographical notes. The chapter headings are: 1. Preliminaries; 2. Linear and affine structures; 3. The commutation matrix; 4. Symmetry - the duplication matrix; 5. Lower triangularity - the elimination matrix; 6. Skew-symmetry and strict lower triangularity; 7. Diagonality and other L-structures; 8. Jacobians; 9. Applications to matrix equations and optimization; 10. Applications to the normal distribution and maximum likelihood. The book gives a good account of an area of linear algebra which is of importance to graduate students and researchers in econometrics and statistics. (Two small errors: the definition of orthonormal in Exercise 1.1 (p. 5) is incomplete and the second \(\Sigma\) in formula 1.1 (p. 6) should be a \(\Pi\).) Reviewer: F.J.Gaines Cited in 2 ReviewsCited in 65 Documents MSC: 15B36 Matrices of integers 15-02 Research exposition (monographs, survey articles) pertaining to linear algebra 05B20 Combinatorial aspects of matrices (incidence, Hadamard, etc.) 15A24 Matrix equations and identities 15A09 Theory of matrix inversion and generalized inverses 62H10 Multivariate distribution of statistics 62H12 Estimation in multivariate analysis 15B57 Hermitian, skew-Hermitian, and related matrices Keywords:linear structure; Toeplitz matrices; 0-1 matrix; commutation matrix; duplication matrix; monograph; matrix equations; normal distribution; maximum likelihood; exercises; Symmetry; Lower triangularity; elimination matrix; Skew-symmetry; Diagonality; L-structures; Jacobians; optimization; econometrics; statistics PDFBibTeX XMLCite \textit{J. R. Magnus}, Linear structures. London: Charles Griffin \&| Company Ltd.; New York etc.: Oxford University Press (1988; Zbl 0667.15010)